W. van Saarloos, "Front propagation into unstable states," Phys. Rep. 386 , 29

W. van Saarloos, "Front propagation into unstable states," Phys. Rep. 386 , 29
Available online at www.sciencedirect.com
Physics Reports 386 (2003) 29 – 222
www.elsevier.com/locate/physrep
Front propagation into unstable states
Wim van Saarloos∗
Instituut-Lorentz, Universiteit Leiden, Postbus 9506, 2300 RA Leiden, The Netherlands
Accepted 12 August 2003
editor: C.W.J. Beenakker
Abstract
This paper is an introductory review of the problem of front propagation into unstable states. Our presentation is centered around the concept of the asymptotic linear spreading velocity v∗ , the asymptotic rate with
which initially localized perturbations spread into an unstable state according to the linear dynamical equations
obtained by linearizing the fully nonlinear equations about the unstable state. This allows us to give a precise
de3nition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v∗ , and pushed fronts,
nonlinear fronts whose asymptotic speed v† is larger than v∗ . In addition, this approach allows us to clarify
many aspects of the front selection problem, the question whether for a given dynamical equation the front
is pulled or pushed. It also is the basis for the universal expressions for the power law rate of approach of
the transient velocity v(t) of a pulled front as it converges toward its asymptotic value v∗ . Almost half of the
paper is devoted to reviewing many experimental and theoretical examples of front propagation into unstable
states from this uni3ed perspective. The paper also includes short sections on the derivation of the universal
power law relaxation behavior of v(t), on the absence of a moving boundary approximation for pulled fronts,
on the relation between so-called global modes and front propagation, and on stochastic fronts.
c 2003 Elsevier B.V. All rights reserved.
PACS: 47.54.+r; 47.20.−k; 02.30.Jr; 82.40.Ck; 47.20.Ky; 87.18.Hf
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1. Scope and aim of the article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2. Motivation: a personal historical perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2. Front propagation into unstable states: the basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1. The linear dynamics: the linear spreading speed v∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. The linear dynamics: characterization of exponential solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. The linear dynamics: importance of initial conditions and transients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
∗
Tel.: +31-71-5275501; fax: +31-71-5275511.
E-mail address: [email protected] (W. van Saarloos).
c 2003 Elsevier B.V. All rights reserved.
0370-1573/$ - see front matter doi:10.1016/j.physrep.2003.08.001
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2.4. The linear dynamics: generalization to more complicated types of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5. The linear dynamics: convective versus absolute instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6. The two-fold way of front propagation into linearly unstable states:
pulled and pushed fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7. Front selection for uniformly translating fronts and coherent and incoherent
pattern forming fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.1. Uniformly translating front solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.2. Coherent pattern forming front solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.3. Incoherent pattern forming front solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.4. EDects of the stability of the state generated by the front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.5. When to expect pushed fronts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7.6. Precise determination of localized initial conditions which give rise to pulled and pushed fronts,
and leading edge dominated dynamics for non-localized initial conditions . . . . . . . . . . . . . . . . . . . . . . .
2.7.7. Complications when there is more than one linear spreading point . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8. Relation with existence and stability of front stability and relation with
previously proposed selection mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.1. Stability versus selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.2. Relation between the multiplicity of front solutions and their stability spectrum . . . . . . . . . . . . . . . . . .
2.8.3. Structural stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8.4. Other observations and conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9. Universal power law relaxation of pulled fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.1. Universal relaxation towards a uniformly translating pulled front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.2. Universal relaxation towards a coherent pattern forming pulled front . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9.3. Universal relaxation towards an incoherent pattern forming pulled front . . . . . . . . . . . . . . . . . . . . . . . . .
2.10. Nonlinear generalization of convective and absolute instability on the basis of the results for front
propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11. Uniformly translating fronts and coherent and incoherent pattern forming fronts in
fourth order equations and CGL amplitude equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.1. The extended Fisher–Kolmogorov equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.2. The Swift–Hohenberg equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.3. The Cahn–Hilliard equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.4. The Kuramoto–Sivashinsky equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.5. The cubic complex Ginzburg–Landau equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.11.6. The quintic complex Ginzburg–Landau equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.12. Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Experimental and theoretical examples of front propagation into unstable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1. Fronts in Taylor–Couette and Rayleigh–BIenard experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. The propagating Rayleigh–Taylor instability in thin 3lms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Pearling, pinching and the propagating Rayleigh instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Spontaneous front formation and chaotic domain structures in rotating Rayleigh–BIenard convection . . . . . . .
3.5. Propagating discharge fronts: streamers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6. Propagating step fronts during debunching of surface steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7. Spinodal decomposition in polymer mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8. Dynamics of a superconducting front invading a normal state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9. Fronts separating laminar and turbulent regions in parallel shear Kows:
Couette and Poiseuille Kow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.10. The convective instability in the wake of bluD bodies: the BIenard–Von Karman vortex street . . . . . . . . . . . .
3.11. Fronts and noise-sustained structures in convective systems I:
the Taylor–Couette system with through Kow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12. Fronts and noise-sustained structures in convective systems II: coherent and incoherent sources and the
heated wire experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.13. Chemical and bacterial growth fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.14. Front or interface dynamics as a test of the order of a phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15. Switching fronts in smectic C ∗ liquid crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.16. Transient patterns in structural phase transitions in solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.17. Spreading of the Mullins–Sekerka instability along a growing interface and the origin of side-branching . . .
3.18. Combustion fronts and fronts in periodic or turbulent media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.19. Biological invasion problems and time delay equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.20. Wound healing as a front propagation problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.21. Fronts in mean 3eld approximations of growth models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.22. Error propagation in extended chaotic systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.23. A clock model for the largest Lyapunov exponent of the particle trajectories in a dilute gas . . . . . . . . . . . . .
3.24. Propagation of a front into an unstable ferromagnetic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.25. Relation with phase transitions in disorder models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26. Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.1. Renormalization of disorder models via traveling waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.2. Singularities and “fronts” in cascade models for turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.3. Other biological problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.4. Solar and stellar activity cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.26.5. Digital search trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4. The mechanism underlying the universal convergence towards v∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1. Two important features of the linear problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. The matching analysis for uniformly translating fronts and coherent pattern forming fronts . . . . . . . . . . . . . . .
4.3. A dynamical argument that also holds for incoherent fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5. Breakdown of moving boundary approximations of pulled fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1. A spherically expanding front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2. Breakdown of singular perturbation theory for a weakly curved pulled front . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3. So what about patterns generated by pulled fronts? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6. Fronts and emergence of “global modes” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1. A front in the presence of an overall convective term and a zero boundary condition
at a 3xed position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Fronts in nonlinear global modes with slowly varying (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7. Elements of stochastic fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1. Pulled fronts as limiting fronts in diDusing particle models: strong cutoD eDects . . . . . . . . . . . . . . . . . . . . . . . .
7.2. Related aspects of Kuctuating fronts in stochastic Langevin equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. The universality class of the scaling properties of Kuctuating fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A. Comparison of the two ways of evaluating the asymptotic linear spreading problem . . . . . . . . . . . . . . . .
Appendix B. Additional observations and conjectures concerning front selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix C. Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction
1.1. Scope and aim of the article
The aim of this article is to introduce, discuss and review the main aspects of front propagation
into an unstable state. By this we mean that we will consider situations in spatio-temporally extended
systems where the (transient) dynamics is dominated by a well-de3ned front which invades a domain
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in which the system is in an unstable state. With the statement that the system in the domain into
which the front propagates is in an unstable state, we mean that the state of the system in the region
far ahead of the front is linearly unstable. In the prototypical case in which this unstable state is a
stationary homogeneous state of the system, this simply means that if one takes an arbitrarily large
domain of the system in this state and analyzes its linear stability in terms of Fourier modes,
a continuous set of these modes is unstable, i.e., grows in time.
At 3rst sight, the subject of front propagation into unstable states might seem to be an esoteric
one. After all, one might think that examples of such behavior would hardly ever occur cleanly
in nature, as they appear to require that the system is 3rst prepared carefully in an unstable state,
either by using special initial conditions in a numerical simulation or by preparing an experimental
system in a state it does not naturally stay in. In reality, however, the subject is not at all of purely
academic interest, as there are many examples where either front propagation into an unstable state is
an essential element of the dynamics, or where it plays an important role in the transient behavior.
There are several reasons for this. First of all, there are important experimental examples where
the system is essentially quenched rapidly into an unstable state. Secondly, fronts naturally arise
in convectively unstable systems, systems in which a state is unstable, but where in the relevant
frame of reference perturbations are convected away faster than they grow out—it is as if in such
systems the unstable state is actually dynamically produced since the convective eDects naturally
sweep the system clean. Even if this is the case in an in3nite system, fronts do play an important
role when the system is 3nite. For example, noise or a perturbation or special boundary condition
near a 3xed inlet can then create patterns dominated by fronts. Moreover, important changes in
the dynamics usually occur when the strength of the instability increases, and the analysis of the
point where the instability changes over from convectively unstable to absolutely unstable (in which
case perturbations in the relevant frame do grow faster than they are convected away) is intimately
connected with the theory of front propagation into unstable states. Thirdly, as we shall explain in
more detail later, close to an instability threshold front propagation always wins over the growth of
bulk modes.
The general goal of our discussion of front propagation into unstable states is to investigate the
following front propagation problem:
If initially a spatially extended system is in an unstable state everywhere except in some
spatially localized region, what will be the large-time dynamical properties and speed of
the nonlinear front which will propagate into the unstable state? Are there classes of initial
conditions for which the front dynamics converges to some unique asymptotic front state?
If so, what characterizes these initial conditions, and what can we say about the asymptotic
front properties and the convergence to them?
Additional questions that may arise concern the sensitivity of the fronts to noise or a 3xed perturbation modeling an experimental boundary condition or an inlet, or the question under what conditions
the fronts can be mapped onto an eDective interface model when they are weakly curved.
Our approach to introducing and reviewing front propagation into unstable states will be based
on the insight that there is a single unifying concept that allows one to approach essentially all
these questions for a large variety of fronts. This concept is actually very simple and intuitively
appealing, and allows one to understand the majority of examples one encounters with just a few
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
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Fig. 1. Graphical summary of one of the major themes of this paper. From top to bottom: linear spreading, pulled
fronts and pushed fronts. From left to right: uniformly translating fronts, coherent pattern forming fronts and incoherent
pattern forming fronts. The plots are based on numerical simulations of three diDerent types of dynamical equations
discussed in this paper. In all cases, the initial condition was a Gaussian of height 0.1, and the state to the right is linearly
unstable. To make the dynamics visible in these space-time plots, successive traces of the fronts have been moved upward.
Thicks along the vertical axes are placed a distance 2.5 apart. √
Left column: F-KPP equation (1) with a pulled front with
f(u) = u − u3 (middle) and a pushed one for f(u) = u + 2 3u3 − u5 in the lower row, for times up to 42. Middle
column: the Swift–Hohenberg equation of Section 2.11.2 (middle) and an extension of it as in Fig. 14(b) (bottom).
Right column: Kuramoto–Sivashinsky equation discussed in Section 2.11.4 (middle) and an extension of it, as in Fig. 16,
but with c = 0:17 (bottom).
related theoretical tools. Its essence can actually be stated in one single sentence:
Associated with any given unstable state is a well-de6ned and easily calculated so-called
“linear” spreading velocity v∗ , the velocity with which arbitrarily small linear perturbations
about the unstable state grow out and spread according to the dynamical equations obtained
by linearizing the full model about the unstable state; nonlinear fronts can either have their
asymptotic speed vas equal to v∗ (a so-called pulled front) or larger than v∗ (a pushed front).
The name pulled front stems from the fact that such a front is, as it were, being “pulled along” by
the leading edge of the front, the region where the dynamics of the front is to good approximation
governed by the equations obtained by linearizing about the unstable state. The natural propagation
speed of the leading edge is hence the asymptotic linear spreading speed v∗ . In this way of thinking,
a pushed front is being pushed from behind by the nonlinear growth in the nonlinear front region
itself [333,334,384].
The fact that the linear spreading velocity is the organizing principle for the problem of front
propagation is illustrated in Fig. 1 for all three classes of fronts, simple uniformly translating fronts,
and coherent and incoherent pattern forming fronts. In the upper panels, we show simulations of the
spreading of an initial perturbation into the unstable state according to the linear equations, obtained
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
by linearizing the model equation about the unstable state. This illustrates the linear spreading
problem associated with the linear dynamics. The asymptotic linear spreading speed v∗ can be
calculated explicitly for any given dynamical equation. Note that since the dynamical equations have
been linearized, there is no saturation: The dynamical 3elds in the upper panels continue to grow
and grow (in the plots in the middle and on the right, the 3eld values also grow to negative values,
but this is masked in such a hidden-line plot). The middle panels show examples of pulled fronts:
These are seen to advance asymptotically with the same speed v∗ as the linear spreading problem
of the upper panel. The lower panels illustrate pushed fronts, whose asymptotic speed is larger than
the linear spreading speed v∗ . The fronts in the left column are uniformly translating, those in
the middle column are coherent pattern forming fronts, and those in the right incoherent pattern
forming fronts. We will de3ne these front classi3cations more precisely later in Section 2.7—for
now it suRces to become aware of the remarkable fact that in spite of the diDerence in appearance
and structure of these fronts, it is useful to divide fronts into two classes, those which propagate
with asymptotic speed v∗ and those whose asymptotic speed v† is larger. Explaining and exploring
the origin and rami3cations of this basic fact is one of the main goals of this article.
In line with our philosophy to convey the power of this simple concept, we will 3rst only present
the essential ingredients that we think a typical non-expert reader should know, and then discuss
a large variety of experimental and theoretical examples of front propagation that can indeed be
understood to a large extent with the amount of theoretical baggage that we equip the reader with
in Section 2. Only then will we turn to a more detailed exposition of some of the more technical
issues underlying the presentation of Section 2, and to a number of advanced topics. Nevertheless,
throughout the paper our philosophy will be to focus on the essential ideas and to refer for the
details to the literature—we will try not to mask the common and unifying features with too many
details and special cases, even though making some caveats along the lines will be unavoidable.
In fact, even in these later chapters, we will see that the above simple insight is the main idea
that also brings together various important recent theoretical developments: the derivation of an
exact results for the universal power law convergence of pulled fronts to their asymptotic speed,
the realization that many of these results extend without major modi3cation to fronts in diDerence
equations or fronts with temporal or spatial kernels, the realization that curved pulled fronts in more
than one dimension cannot be described by a moving boundary approximation or eDective interface
description, as well as the eDects of a particle cutoD on fronts, and the eDects of Kuctuations.
A word about referencing: when referring to several papers in one citation, we will do so in the
numerical order imposed by the alphabetic reference list, not in order of importance of the references.
If we want to distinguish papers, we will reference them separately.
1.2. Motivation: a personal historical perspective
My choice to present the theory this way is admittedly very personal and unconventional, but is
made deliberately. The theory of front propagation has had a long and twisted but interesting history,
with essential contributions coming from diDerent directions. I feel it is time to take stock. The 3eld
started essentially some 65 years ago 1 with the work of Fisher [163] and KolmogoroD, Petrovsky,
1
As mentioned by Murray [311, p. 277], the Fisher equation was apparently already considered in 1906 by Luther,
who obtained the same analytical form as Fisher for the wave front.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
35
and PiscounoD [234] on fronts in nonlinear diDusion type equations motivated by population dynamics issues. The subject seems to have remained mainly in mathematics initially, culminating in the
classic work of Aronson and Weinberger [15,16] which contains a rather complete set of results for
the nonlinear diDusion equation (a diDusion equation for a single variable with a nonlinear growth
term, Eq. (1) below). The special feature of the nonlinear diDusion equation that makes most of
the rigorous work on this equation possible is the existence of a so-called comparison theorem,
which allows one to bound the actual solution of the nonlinear diDusion equation by suitably chosen
simpler ones with known properties. Such an approach is mathematically powerful, but is essentially
limited by its nature to the nonlinear diDusion equation and its extensions: A comparison theorem
basically only holds for the nonlinear diDusion equation or variants thereof, not for the typical types
of equations that we encounter in practice and that exhibit front propagation into an unstable state
in a pattern forming system.
In the early 1980s of the last century, the problem of front propagation was brought to the attention
of physicists by Langer and coworkers [38,111,248], who noted that there are some similarities between what we will call the regime of pulled front propagation and the (then popular) conjecture that
the natural operating point of dendritic growth was the “marginally stable” front solution [247,248],
i.e., the particular front solution for which the least stable stability eigenmode changes from stable
to unstable (for dendrites, this conjecture was later abandoned). In addition, they re-interpreted the
two modes of operation 2 of front dynamics in terms of the stability of front solutions [38]. This
point of view also brought to the foreground the idea that front propagation into unstable states
should be thought of as an example of pattern selection: since there generally exists a continuum
of front solutions, the question becomes which one of these is “selected” dynamically for a large
class of initial conditions. For this reason, much of the work in the physics community following
this observation was focused on understanding this apparent connection between the stability of front
pro3les and the dynamical selection mechanism [83,333,334,354,380,420,421]. Also in my own work
along these lines [420,421] I pushed various of the arguments for the connection between stability and selection. This line of approach showed indeed that the two regimes of front propagation
that were already apparent from the work on the nonlinear diDusion equation do in fact have their
counterparts for pattern forming fronts, fronts which leave a well-de3ned 3nite-wavelength pattern
behind. In addition, it showed that the power law convergence to the asymptotic speed known for
the nonlinear diDusion equation [54] is just a speci3c example of a generic property of fronts in the
“linear marginal stability” [420,421] regime—the “pulled” regime as we will call it here. Nevertheless, although some of these arguments have actually made it into a review [105] and into textbooks
[189,320], they remain at best a plausible set of arguments, not a real theoretical framework; this is
illustrated by the fact that the term “marginal stability conjecture” is still often used in the literature,
especially when the author seems to want to underline its somewhat mysterious character.
Quite naturally, the starting point of the above line of research was the nonlinear evolution of
fronts solutions. From this perspective it is understandable that many researchers were intrigued
but also surprised to see that in the pulled (or linear marginal stability) front regime almost all the
essential properties of the fronts are determined by the dispersion relation of the linearized dynamics
of arbitrarily small perturbations about the unstable state. Perhaps this also explains, on hindsight,
why for over 30 years there was a virtually independent line of research that originated in plasma
2
Their “case I” and “case II” [38] are examples of what we refer to here as pulled and pushed front solutions.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
physics and Kuid dynamics. In these 3elds, it is very common that even though a system is linearly
unstable (in other words, that when linearized about a homogeneous state, there is a continuous
range of unstable Fourier modes), it is only convectively unstable. As mentioned before, this means
that in the relevant frame a localized perturbation is convected away faster than it is growing out.
To determine whether a system is either convectively unstable or absolutely unstable mathematically
translates into studying the long-time asymptotics of the Green’s function of the dynamical equations,
linearized about the unstable state. 3 The technique to do so was developed in the 1950s [62] and is
even treated in one of the volumes of the Landau and Lifshitz course on theoretical physics [264],
but appears to have gone unnoticed in the mathematics literature. It usually goes by the name of
“pinch point analysis” [49,204,205]. As we will discuss, for simple systems it amounts to a saddle
point analysis of the asymptotics of the Green’s function. In 1989 I pointed out [421] that the
equations for the linear spreading velocity of perturbations, according to this analysis, the velocity
we will refer to as v∗ , are actually the same as the expressions for the speed in the “linear marginal
stability” regime of nonlinear front propagation [38,111,421]. Clearly, this cannot be a coincidence,
but the general implications of this observation appear not to have been explored for several more
years. One immediate simple but powerful consequence of this connection is that it shows that the
concept of the linear spreading velocity v∗ applies equally well to diDerence equations in space
and time—after all in Fourier language, in which the asymptotic analysis of the Green’s function
analysis is most easily done, putting a system on a lattice just means that the Fourier integrals are
restricted to a 3nite range (a physicist would say: restricted to the Brillouin zone). The concept of
linear spreading velocity also allows one to connect front propagation with work in recent years on
the concept of global modes in weakly inhomogeneous unstable systems [98,99].
Most of the work summarized above was con3ned to fronts in one dimension. The natural approach to analyze nontrivial patterns in more dimensions on scales much larger than the typical front
width is, of course, to view the front on the large pattern scale as a sharp moving interface—in
technical terms, this means that one would like to apply singular perturbation theory to derive a
moving boundary approximation or an eDective interface approximation (much like what is often
done for the so-called phase-3eld models that have recently become popular [29,71,219]). When
this was attempted for discharge patterns whose dynamics is governed by “pulled” fronts [141,142],
the standard derivation of a moving boundary approximation was found to break down. Mathematically, this traced back to the fact that for pulled fronts the dynamically important region is
ahead of the nonlinear transition zone which one normally associates with the front itself. This was
another important sign that one really has to take the dynamics in the region ahead of the front
seriously!
My view that the linear spreading velocity is the proper starting point for understanding the two
regimes of front propagation into unstable states, and for tying together the various theoretical developments and experiments—and hence that an introductory review should be organized around it—is
colored by the developments sketched above and in particular the fact that Ebert and I have recently
3
Some readers may be amused to note that there are traces of such arguments in the original paper of Kolmogorov
et al. [234]: although hidden, the Green’s function of the diDusion equation plays an important role in their convergence
proofs. This makes me believe that it is likely that it will be possible to prove results concerning front propagation into
unstable states for more complicated equations like higher order partial diDerential equations, by putting bounds on the
Green’s function.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
37
been able to derive from it important new and exact results for the power law convergence of the
velocity and shape of a pulled front to its asymptotic value [143,144]. The fact that starting from
this concept one can set up a fully explicit calculational scheme to study the long time power law
convergence or relaxation and that this yields new universal terms which are exact (and which even
for the nonlinear diDusion equation [15,16] go beyond those which were previously known [54]),
shows more than anything else that we have moved from the stage of speculations and intuitive
concepts to what has essentially become a well-de3ned and powerful theoretical framework. My
whole presentation builds on the picture coming out of this work [143–147].
As mentioned before, the subject of front propagation also has a long history in the mathematics
literature; moreover, especially in the last 10 –15 years a lot of work has been done on coherent
structure type solutions like traveling fronts, pulses etc. With such a diverse 3eld, spread throughout
many disciplines, one cannot hope to do justice to all these developments. My choice to approach
the subject from the point of view of a physicist just reKects that I only feel competent to review
the developments in this part of the 3eld, and that I do want to open up the many advances that
have been made recently to researchers with diDerent backgrounds who typically will not scan the
physics literature. I will try to give a fair assessment of some of the more mathematical developments
but there is absolutely no claim to be exhaustive in that regard. Luckily, authoritative reviews of
the more recent mathematical literature are available [170,172,434,442]. The second reason for my
choice is indeed that most of the mathematics literature is focused on equations that admit uniformly
translating front solutions. For many pattern forming systems, the relevant front solutions are not
of this type, they are either coherent or incoherent pattern forming fronts of the type we already
encountered in Fig. 1 (these concepts are de3ned precisely in Section 2.7). Even though not much
is known rigorously about these more general pattern forming fronts, our presentation will allow us
to approach all types of fronts in a uni3ed way that illuminates what is and is not known. We hope
this will also stimulate the more mathematically inclined reader to take up the challenge of entering
an area where we do know most answers but lack almost any proof. I am convinced a gold mine
is waiting for those who dare.
As explained above, we will 3rst introduce in Section 2 the key ingredients of front propagation
into unstable states that provide the basic working knowledge for the non-expert physicist. The
introduction along this line also allows us to identify most clearly the open problems. We then turn
right away to a discussion of a large number of examples of front propagation. After this, we will
give a more detailed discussion of the slow convergence of pulled fronts to their asymptotic velocity
and shape. We are then in a position to discuss what patterns, whose dynamics is dominated by fronts
propagating into an unstable state, can be analyzed in terms of a moving boundary approximation, in
the limit that the front is thin compared to the pattern scale. This is followed by a discussion of the
relation with the existence of “global modes” and of some of the issues related to stochastic fronts.
2. Front propagation into unstable states: the basics
The central theme of this paper is to study fronts in spatio-temporally extended systems which
propagate into a linearly unstable state. The special character and, to a large extent, simplicity of
such fronts arises from the fact that arbitrarily small perturbations about the unstable state already
grow and spread by themselves, and thus have a nontrivial—and, as we shall see, rather universal—
dynamics by themselves. The “linear spreading speed” v∗ with which small perturbations spread out
38
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
is then automatically an important reference point. This is diDerent from fronts which separate two
linearly stable states—in that case the perturbations about each individual stable state just damp out
and there is not much to be gained from studying precisely how this happens; instead, the motion
of such fronts is inherently nonlinear. 4
It will often be instructive to illustrate our analysis and arguments with a simple explicit example;
to this end we will use the famous nonlinear diDusion equation with which the 3eld started,
9t u(x; t) = 92x u(x; t) + f(u);
with
f(0) = 0; f(1) = 0 ;
f (0) = 1; f (1) ¡ 0 :
(1)
This is the equation studied by Fisher [163] and Kolmogorov et al. [234] back in 1937, and we shall
therefore follow the convention to refer to it as the F-KPP equation. As we mentioned already in the
introduction, this equation and its extensions have been the main focus of (rigorous) mathematical
studies of front propagation into unstable states, but these are not the main focus of this review—
rather, we will use the F-KPP equation only as the simplest equation to illustrate the points which are
generic to the front propagation problem, and will not rely on comparison-type methods or bounds
which are special to this equation. 5 At this point it simply suRces to note that the state u = 0 of the
real 3eld u is an unstable state: when u is positive but small, f(u) ≈ f (0)u = u, so the second term
on the right-hand side of the F-KPP equation drives u away from zero. The front propagation problem
we are interested in was already illustrated in Fig. 1: We want to determine the long time asymptotic
behavior of the front which propagates to the right into the unstable state u=0, given initial conditions
for which u(x → ∞; t = 0) = 0. A simple analysis based on constructing the uniformly translating
front solutions u(x − vt) does not suRce, as there is a continuous family of such front solutions.
Since the argument can be found at many places in the literature [15,38,105,144,249,268,421,428],
we will not repeat it here.
2.1. The linear dynamics: the linear spreading speed v∗
Our approach to the problem via the introduction of the linear spreading speed v∗ is a slight
reformulation of the analysis developed over 40 years ago in plasma physics [49,62,264]. We 3rst
formulate the essential concept having in mind a simple partial diDerential equation or a set of partial
diDerential equations, and then brieKy discuss the minor complications that more general classes of
dynamical equations entail. We postpone the discussion of fronts in higher dimensions to Section 5,
so we limit the discussion here to fronts in one dimension.
Suppose we have a dynamical problem for some 3eld, which we will generically denote by (x; t),
whose dynamical equation is translation invariant and has a homogeneous stationary state =0 which
is linearly unstable. With this we mean that if we linearize the dynamical equation in about the
unstable state, then Fourier modes grow for some range of spatial wavenumbers k. More concretely,
4
Technically, determining the asymptotic fronts speed then usually amounts to a nonlinear eigenvalue problem. The
spreading of the precursors of such fronts is studied in [227].
5
A physicist’s introduction to comparison type arguments can be found in the appendix of [38]; for recent work on
bounds on the asymptotic speed of fronts in the F-KPP equation see in particular [33–37]. A new look at the global
variational structure was recently given in [310]. A recent example of a convergence proof in coupled equations can be
found in [323].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
39
Fig. 2. Qualitative sketch of the growth and spreading of the 3eld (x; t) according to the dynamical evolution equation
linearized about the unstable state = 0. The successive curves illustrate the initial condition (x; t0 ) and the 3eld (x; t)
at successive times t1 ¡ t2 ¡ t3 ¡ t4 . Note that there is obviously no saturation of the 3eld in the linearized dynamics:
The asymptotic spreading velocity v∗ to the right is the asymptotic speed of the positions xC (t) where (x; t) reaches the
level line = C: (xC (t); t) = C. The asymptotic spreading velocity to the left is de3ned analogously.
if we take a spatial Fourier transform and write
∞
˜
(k; t) =
d x (x; t)e−ikx ;
−∞
(2)
substitution of the Ansatz
−i!(k)t
˜ t) = (k)e
V
(k;
(3)
yields the dispersion relation !(k) of Fourier modes of the linearized equation. We will discuss the
situation in which the dispersion relation has more than one branch of solutions later, and for now
assume that it just has a single branch. Then the statement that the state = 0 is linearly unstable
simply means that
= 0 linearly unstable : Im !(k) ¿ 0 for some range of k-values :
(4)
At this stage, the particular equation we are studying is simply encoded in the dispersion relation
!(k). 6 This dispersion relation can be quite general—we will come back to the conditions on !(k)
in Section 2.4 below, and for now will simply assume that !(k) is an analytic function of k in the
complex k-plane.
We are interested in studying the long-time dynamics emerging from some generic initial condition
which is suRciently localized in space (we will make the term “suRciently localized” more precise
in Section 2.3 below). Because there is a range of unstable modes which grow exponentially in time
as eIm !(k)t , a typical localized initial condition will grow out and spread in time under the linear
dynamics as sketched in Fig. 2. If we now trace the level curve xC (t) where (xC (t); t) = C in
space-time, as indicated in the 3gure, the linear spreading speed v∗ is de3ned as the asymptotic
6
The reader who may prefer to see an example of a dispersion relation is invited to check the dispersion relation (86)
for Eq. (85).
40
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
speed of the point xC (t):
d xC (t)
v∗ ≡ lim
:
(5)
t →∞
dt
Note that v∗ is independent of the value of C because of the linearity of the evolution equation.
However, for systems whose dynamical equations are not reKection symmetric, as happens quite
often in Kuid dynamics and plasmas, one does have to distinguish between a spreading speed to the
left and one to the right. In order not to overburden our notation, we will in this paper by convention
always focus on the spreading velocity of the right =ank of ; this velocity is counted positive if
this Kank spreads to the right, and negative when it recedes to the left.
V
Given !(k) and (k),
which according to (3) is just the Fourier transform of the initial condition
(x; t = 0), one can write (x; t) for t ¿ 0 simply as the inverse Fourier transform
∞
1
ikx−i!(k)t
V
(x; t) =
d k (k)e
:
(6)
2 −∞
The more general Green’s function formulation will be discussed later in Section 2.4. Our de3nition
of the linear spreading speed v∗ to the right is illustrated in Fig. 2. We will work under the assumption
that the asymptotic spreading speed v∗ is 3nite; whether this is true can always be veri3ed selfconsistently at the end of the calculation. The existence of a 3nite v∗ implies that if we look in frame
= x − v∗ t
(7)
moving with this speed, we neither see the right Kank grow nor decay exponentially. To determine
v∗ , we therefore 3rst write the inverse Fourier formula (6) for in this frame,
∞
1
ik(x−v∗ t)−i[!(k)−v∗ k]t
V
d k (k)e
;
(; t) =
2 −∞
∞
1
ik−i[!(k)−v∗ k]t
V
d k (k)e
;
(8)
=
2 −∞
and then determine v∗ self-consistently by analyzing when this expression neither leads to exponential growth nor to decay in the limit 6nite, t → ∞. We cannot simply evaluate the integral by
closing the contour in the upper half of the k-plane, since the large-k behavior of the exponent is
normally dominated by the large-k behavior of !(k). However, the large-time limit clearly calls for a
saddle-point approximation [32] (also known as stationary phase or steepest descent approximation):
Since t becomes arbitrarily large, we deform the k-contour to go through the point in the complex k
plane where the term between square brackets varies least with k, and the integral is then dominated
by the contribution from the region near this point. This so-called saddle point k ∗ is given by
d!(k) d[!(k) − v∗ k] ∗
(9)
∗ = 0 ⇒ v = dk ∗ :
dk
k
k
These saddle point equations will in general have solutions in both the upper and the lower half of
the complex k-plane; the ones in the upper half correspond to the asymptotic decay towards large
x in the Fourier decomposition (8) associated with the right Kank of the perturbation sketched in
Fig. 2, and those in the lower half to an exponentially growing solution for increasing x and thus to
the behavior on the left Kank. By convention, we will focus on the right Kank, which may invade
the unstable state to the right. If we deform the k-contour into the complex plane to go through the
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
41
V
saddle point in the upper half plane, and assume for the moment that (k),
the Fourier transform of
the initial condition, is an entire function (one that is analytic in every 3nite region of the complex
k-plane), the dominant term to the integral is nothing but the exponential factor in (8) evaluated
∗
∗ ∗
at the saddle-point, i.e., ei[!(k )−v k ]t . The self-consistency requirement that this term neither grows
nor decays exponentially thus simply leads to
Im !(k ∗ ) !i
Im !(k ∗ ) − v∗ Im k ∗ = 0 ⇒ v∗ =
=
:
(10)
Im k ∗
ki
The notation !i which we have introduced here for the imaginary part of ! will be used interchangeably from now on with Im !; likewise, we will introduce the subindex r to denote the real
part of a complex quantity. Upon expanding the factor in the exponent in (8) around the saddle
point value given by Eqs. (9) and (10), we then get from the resulting Gaussian integral
∞
1
−i!r∗ t+i(k ∗ +Wk)−Dt(Wk)2
V
(; t) d k (k)e
(Wk = k − k ∗ ) ;
2 −∞
1 ik ∗ −(!r∗ −kr v)t ∞
−Dt[Wk −i=2Dt]2 −2 =4Dt
V
e
d k (k)e
;
2
−∞
1
∗
∗
2
V ∗ ) ( 3xed; t → ∞) ;
eik −i!r t e− =4Dt (k
(11)
4Dt
where all parameters are determined by the dispersion relation through the saddle point values,
!i (k ∗ )
!i (k ∗ )
i d 2 !(k) d!(k) ∗
=
; v =
; D=
:
(12)
d k k ∗
ki∗
ki∗
2 d k 2 k ∗
√
Since ! and k are in general complex, the 3rst equation can actually be thought of as two equations
for the real and imaginary parts, which can be used to solve for k ∗ . The second and third equation
then give v∗ and D.
∗
The exponential factor eik gives the dominant spatial behavior of in the co-moving frame on
the right Kank in Fig. 2: if we de3ne the asymptotic spatial decay rate ∗ and the eDective di>usion
coe?cient 7 D by
1
1
∗ ≡ Im k ∗ ;
≡ Re
;
(13)
D
D
then we see that the modulus of falls oD as
1
∗
2
(14)
|(; t)| ∼ √ e− e− =4Dt ( 3xed; t → ∞) ;
t
i.e., essentially as e− with a Gaussian correction that is reminiscent of diDusion-like behavior.
We will prefer not to name the point k ∗ after the way it arises mathematically (e.g., saddle point
or “pinch point”, following the formulation discussed in Section 2.4). Instead, we will usually refer
to k ∗ as the linear spreading point; likewise, expressions (11) and (14) for will be referred to as
the linear spreading pro3les.
∗
7
We stress that D is the eDective diDusion coeRcient associated with the saddle point governing the linear spreading
behavior of the deterministic equation. In Section 7 we will also encounter a front diDusion coeRcient Dfront which is a
measure for the stochastic front wandering, but this is an entirely diDerent quantity.
42
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
For an ordinary diDusion process to be stable, the diDusion coeRcient has to be positive. Thus we
expect that in the present case D should be positive. Indeed, the requirement that the linear spreading point corresponds to a maximum of the exponential term in (8) translates into the condition,
ReD¿0, and this entails D¿0. We will come back to this and other conditions in Section 2.4 below.
In spite of the simplicity of their derivation and form, Eqs. (11) and (12) express the crucial
result that as we shall see permeates the 3eld of front propagation into unstable states:
associated with any linearly unstable state is a linear spreading speed v∗ given by (12); this is
the natural asymptotic spreading speed with which small “su?ciently localized” perturbations
spread into a domain of the unstable state according to the linearized dynamics.
Before turning to the implications for front propagation, we will in the next sections discuss various aspects and generalizations of these concepts, including the precise condition under which
“suRciently localized” initial conditions do lead to an asymptotic spreading velocity v∗
(the so-called steep initial conditions given in (37) below).
Example. Application to the linear F-KPP equation.
Let us close this section by applying the above formalism to the F-KPP equation (1). Upon
linearizing the equation in u, we obtain
linearized F-KPP:
9t u(x; t) = 92x u(x; t) + u :
(15)
Substitution of a Fourier mode e−i!t+ikx gives the dispersion relation
F-KPP:
!(k) = i(1 − k 2 ) ;
(16)
and from this we immediately obtain from (12) and (13)
F-KPP:
∗
= 2;
vFKPP
∗ = 1;
Re k ∗ = 0;
D=D=1 :
(17)
The special simplicity of the F-KPP equation lies in the fact that !(k) is quadratic in k. This not
only implies that the eDective diDusion coeRcient D is nothing but the diDusion coeRcient entering
the F-KPP equation, but also that the exponent in (8) is in fact a Gaussian form without higher
order corrections. Thus, the above evaluation of the integral is actually exact in this case. Another
instructive way to see this is to note that the transformation u = et n transforms the linearized F-KPP
equation (15) into the diDusion equation 9t n = 92x n for n. The fundamental solution corresponding
to delta-function initial condition is the well-known Gaussian; in terms of u this yields
1
2
F-KPP: u(x; t) = √
(18)
et −x =4t (delta function initial conditions) :
4t
2.2. The linear dynamics: characterization of exponential solutions
In the above analysis, we focused immediately on the importance of the linear spreading point
k ∗ of the dispersion relation !(k) in determining the spreading velocity v∗ . Let us now pay more
attention to the precise initial conditions for which this concept is important.
In the derivation of the linear spreading velocity v∗ , we took the Fourier transform of the initial
conditions to be an entire function, i.e., a function which is analytic in any 3nite region of the
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
43
V
complex plane. Thus, the analysis applies to the case in which (x; t = 0) is a delta function ((k)
8
is then k-independent ), has compact support (meaning that (x; t = 0) = 0 outside some 3nite
interval of x), or falls oD faster than any exponential for large enough x (like, e.g., a Gaussian).
V
For all practical purposes, the only really relevant case in which (k)
is not an entire function is
9
when it has poles oD the real axis in the complex plane. This corresponds to an initial condition
(x; t = 0) which falls oD exponentially for large x. To be concrete, let us consider the case in which
V
(k)
has a pole in the upper half plane at k = k . If we deform the k-integral to also go around this
pole, (x; t) also picks up a contribution whose modulus is proportional to 10
|e−i!(k )t+ik x | = e−(x−v(k )t) ;
with ≡ Im k ;
(19)
and whose envelope velocity v(k ) is given by
Im !(k )
:
(20)
Im k We 3rst characterize these solutions in some detail, and then investigate their relevance for the full
dynamics.
Following [144], we will refer to the exponential decay rate of our dynamical 3eld as the
steepness. For a given steepness , !(k ) of course still depends on the real part of k . We choose
to introduce a unique envelope velocity venv () by taking for Re k the value that maximizes Im !
and hence v(k ),
!i (k) 9!i (k) d! venv ( ≡ ki ) =
with
= Im
=0 ;
(21)
ki 9kr dk v(k ) =
k=k
k=k
k=k
where the second condition determines kr implicitly as a function of = ki . The rationale to focus
on this particular velocity as a function of is twofold: First of all, if we consider for the fully
linear problem under investigation here an initial condition whose modulus falls of as e−x but in
whose spectral decomposition a whole range of values of kr are present, this maximal growth value
will dominate the large time dynamics. Secondly, in line with this, when we consider nonlinear front
solutions corresponding to diDerent values of kr , the one not corresponding to the maximum of !i
are unstable and therefore dynamically irrelevant—see Section 2.8.2. Thus, for all practical purposes
the branch of velocities venv () is the real important one.
The generic behavior of venv () as a function of is sketched in Fig. 3(a). In this 3gure, the
dotted lines indicate branches not corresponding to the envelope velocity given by (21): For a given
value of , the other branches correspond to a smaller value of !i and hence to a smaller value of
8
Most of the original literature [49,62,204,264] in which the asymptotic large-time spreading behavior of a perturbation
is obtained through a similar analysis or the more general “pinch point” analysis, is implicitly focused on this case of
delta-function initial conditions, since the analysis is based on a large-time asymptotic analysis of the Greens function of
the dynamical equations. Note in this connection that (18) is indeed the Green’s function solution of the linearized F-KPP
equation.
9
Of course, one may consider other examples of non-analytic behavior, such as power law singularities at k = 0. This
would correspond to a power law initial conditions (x; t = 0) ∼ x− as x → ∞. Such initial conditions are so slowly
decaying that they give an in3nite spreading speed, as (x; t) ∼ eIm !(0)t x− . Also the full nonlinear front solutions have
a divergent speed in this case [256].
10
We are admittedly somewhat cavalier here; a more precise analysis of the crossover between the various contributions
is given in the next subsection below Eq. (31).
44
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 3. (a) Generic behavior of the velocity v() as a function of the spatial decay rate . The thick full line and the
thick dashed line indicate the envelope velocity de3ned in (21): for a given this corresponds to the largest value of
!i and hence to the largest velocity on these branches. The minimum of venv is equal to the linear spreading speed v∗ .
(b) The situation in the special case of uniformly translating solutions which obey !=k = v. The dotted line marks the
branch of solutions with velocity less than v∗ given in (27).
v(). Furthermore, since we are considering the spreading and propagation dynamics at a linearly
unstable state, the maximal growth rate !i () ¿ 0 as ↓ 0. Hence venv () diverges as 1= for → 0.
When we follow this branch for increasing values of , at some point this branch of solutions will
have a minimum. This minimum is nothing but the value v∗ : Since along this branch of solutions
9!i =9kr = 0, we simply have
dvenv
1 9!i !i
1 9!i 9!i d kr
!i
=
;
(22)
=
+
−
−
d
9
9kr d
9
and so at the linear spreading point k ∗
dvenv 1 9!i !i∗
=
− ∗ =0 ;
d k ∗ ∗
9 k ∗
(23)
since at the point k ∗ the term between brackets precisely vanishes, see Eq. (12). By diDerentiating
once more, we see that the curvature of venv () at the minimum can be related to the eDective
diDusion coeRcient 11 D introduced in (13),
d kr 2
1 92 !i 92 !i d kr 92 !i d 2 venv () =
+2
+
d2 ∗ ∗ 92 k ∗
99kr k ∗ d k ∗
9kr2 k ∗ d k ∗
11
Aside for the reader familiar with amplitude equations [105,155,189]: The relation between the curvature of venv ()
at the minimum and the diDusion coeRcient D bears some intriguing similarities to the relation between the curvature of
the growth rate as a function of k of the pattern forming mode near the bifurcation to a 3nite-wavelength pattern and
the parameters in the amplitude equation [105,189,193,316,320]. That curvature is also essentially the diDusion constant
that enters the amplitude equation. Nevertheless, one should keep in mind that the minimum of venv () is associated with
the saddle point of an invasion mode which falls oD in space, not with the maximum growth rate of a Fourier mode.
Moreover, while the amplitude equations only describe pattern formation near the instability threshold, the pulled front
propagation mechanism can be operative far above an instability threshold as well as in pattern forming problems which
have no obvious threshold.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
45
2 2
Di
Di
2
Di2
= ∗ Dr +
− Dr
;
= ∗ Dr + 2Di
Dr
Dr
Dr
2D
;
(24)
∗
where D was de3ned in (12) and where we used the fact that according to the de3nition (13) of D,
we can write D = Dr + Di2 =Dr . Furthermore, in deriving these results, we have repeatedly used the
Cauchy–Riemann relations for complex analytic functions that relate the various derivatives of the
real and imaginary part, and the fact that along the branch of solutions venv , the relation 9!i =9kr = 0
implies Di − Dr (d kr =d) = 0.
If we investigate a dynamical equation which admits a uniformly translating front solution of the
form (x − vt), the previous results need to be consistent which this special type of asymptotic
behavior. Now, the exponential leading edge behavior eikx−i!t we found above only corresponds to
uniformly translating behavior provided
!(k)
( = ki ) :
uniformly translating solutions : v() =
(25)
k
The real part of this equation is consistent with the earlier condition v = !i =ki that holds for all
fronts, but for uniformly translating fronts it implies that in addition Im(!=k) = 0.
Hence, the above discussion is only self-consistent for uniformly translating solutions if the branch
venv () where the growth rate !i is maximal for a given coincides with the condition (25). In all
the cases that I know of, 12 the branch of envelope solutions for v ¿ v∗ coincides with uniformly
translating solutions because the dispersion relation is such that the growth rate !i is maximized for
kr = 0:
=
uniformly translating solutions with v ¿ v∗ : kr = !r = 0;
Di = 0 :
(26)
Obviously, in this case the branch venv () corresponds to the simple exponential behavior exp(−x +
!i t) which is neither temporally nor spatially oscillatory. 13
We had already seen that there generally are also solutions with velocity v ¡ v∗ , as the branches
with velocity venv ¿ v∗ shown in Fig. 3(a) are only those corresponding to the maximum growth
condition 9!i =9kr = 0, see Eq. (21). It is important to realize that if an equation admits uniformly
translating solutions, there is in general also a branch of uniformly translating solutions with v ¡ v∗ .
Indeed, by expanding the curve venv () around the minimum v∗ and looking for solutions with
v ¡ v∗ , one 3nds that these are given by 14
v
(v − v∗ ); kr − kr∗ ≈ 2|v − v∗ |=v (v ¡ v∗ ) :
(27)
− ∗ ≈
2
3(v )
12
As we shall see in Section 2.11.1, the EFK equation illustrates that when the linear spreading point ceases to obey
(26), the pulled fronts change from uniformly translating to coherent pattern forming solutions.
13
For uniformly translating fronts, it would be more appropriate to use in the case of uniformly translating fronts the
usual Laplace transform variables s = −i! and = −ik as these then take real values. We will refrain from doing so since
most of the literature on the asymptotic analysis of the Green’s function on which the distinction between convectively
and absolutely unstable states is built, employs the !-k convention.
14
Note that the formula given on p. 53 of [144] is slightly in error.
46
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
The situation in the special case of uniformly translating solutions is sketched in Fig. 3(b); in this
3gure, the dotted line shows the branch of solutions with v ¡ v∗ . Since solutions for v ¡ v∗ are
always spatially oscillatory (kr = 0), they are sometimes disregarded in the analysis of fronts for
which the dynamical variable, e.g. a particle density, is by de3nition non-negative. It is important
to keep in mind, however, that they do actually have relevance as intermediate asymptotic solutions
during the transient dynamics: as we shall see in Section 2.9, the asymptotic velocity v∗ is always
approached slowly from below, and as a result the transient dynamics follows front solutions with
v ¡ v∗ adiabatically. Secondly, this branch of solutions also pops up in the analysis of fronts in the
case that there is a small cutoD in the growth function—see Section 7.1.
The importance of this simple connection between the minimum of the curve venv () and the
linear spreading speed v∗ can hardly be overemphasized:
For equations of F-KPP type, the special signi6cance of the minimum of the venv () branch as
the selected asymptotic velocity in the pulled regime is well known, and it plays a crucial role
in more rigorous comparison-type arguments for front selection in such types of equations.
The line of argument that we follow here emphasizes that v∗ is the asymptotic speed that
naturally arises from the linearized dynamical problem, and that this is the proper starting
point both to understand the selection problem, and to analyze the rate of convergence to v∗
quantitatively.
Example. Application to the linear F-KPP equation.
We already gave the dispersion relation of the F-KPP equation in (16); using this in Eq. (21)
immediately gives for the upper branches with venv ¿ v∗ = 2
2 −4
venv ± venv
⇔ venv = + −1 ;
F-KPP : =
(28)
2
and for the branches below v∗
1
F-KPP : = v=2; kr = ±
4 − v2 (v ¡ v∗ = 2) ;
(29)
2
in agreement with the above discussion and with (27).
2.3. The linear dynamics: importance of initial conditions and transients
We now study the dependence on initial conditions and the transient behavior. This question is
obviously relevant: The discussion in the previous section shows that simple exponentially decaying
solutions can propagate faster than v∗ —at 3rst sight, one might wonder how a pro3le spreading with
velocity v∗ can ever emerge from the dynamics if solutions exist which tend to propagate faster.
Moreover, as we shall see, initial conditions which fall with an exponential decay rate ¡ ∗ do
give rise to a propagation speed venv () which is larger than v∗ .
If the initial condition is a delta function, or, more generally, if the initial condition has compact
V
support (i.e. vanishes identically outside some 3nite range of x), then the Fourier transform (k)
is
V
an entire function. This means that (k)
is analytic everywhere in the complex k-plane. The earlier
analysis shows that whatever the precise initial conditions, the asymptotic spreading speed is always
simply v∗ , determined by the saddle point of the exponential term.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
47
The only relevant initial conditions which can give rise to spreading with a constant 3nite speed
are the exponential initial conditions already discussed in some detail in the previous subsection.
V
Let us assume that (k)
has a pole in the complex k-plane at k , with spatial decay rate ki = . In
our 3rst round of the discussion, we analyzed the limit 3xed, t → ∞, but it is important to keep in
mind that the limits 3xed, t → ∞ and t 3xed, → ∞ do not commute. Indeed, it follows directly
from the inverse Fourier formula that the spatial asymptotic behavior as x → ∞ is the same as that
of the initial conditions, 15
(x → ∞; t = 0) ∼ e−x ⇒ (x → ∞; t) ∼ e−x :
(30)
In order to understand the competition and crossover between such exponential parts and the contribution from the saddle point, let us return to the intermediate expression (11) that arises in analyzing
the large-time asymptotics,
1 ik ∗ −i!r∗ t ∞
−Dt[Wk −i=2Dt]2 −2 =4Dt
V
(; t) d k (k)e
;
(31)
e
2
−∞
V
and analyze this integral more carefully in a case in which (k)
has a pole whose strength is small.
The term −i=2Dt in the above expression gives a shift in the value of the k where the quadratic
term vanishes. For 3xed , this shift is very small for large t, and the Gaussian integration yields
the asymptotic result (11). However, when is large enough that the point where the growth rate is
maximal moves close to the pole, the saddle point approximation to the integral breaks down. This
clearly happens when the term between brackets in the exponential in (31) is small at the pole, i.e.,
at the crossover point co for which
1
∼ ( − ∗ ) ⇒ co ∼ 2D( − ∗ )t ;
(32)
co Re
2Dt
where we used the eDective diDusion coeRcient D de3ned in (13). This rough argument relates the
velocity and direction of motion of the crossover point to the diDerence in steepness of the initial
condition and the steepness ∗ , and gives insight into how the contributions from the initial condition
and the saddle point dominate in diDerent regions. Before we will discuss this, it is instructive to
give a more direct derivation of a formula for the velocity of the crossover region by matching the
expressions for the 3eld in the two regions. Indeed, the expression for in the region dominated
by the saddle point is the one given in (14),
1
∗
2
V ∗ )| ;
|(; t)| √
e− e− =4Dt |(k
(33)
4Dt
while in the large region the pro3le is simply exponential: in the frame moving with the linear
spreading speed v∗ the pro3le is according to (19)
|(; t)| Ae−[−(venv ()−v
∗
)t]
;
(34)
where A is the pole strength of the initial condition. The crossover point is simply the point where
the two above expression match; by equating the two exponential factors and writing co = vco t, we
obtain from the dominant terms linear in t
2
=4D = −vco + [venv () − v∗ ] ;
− ∗ vco − vco
15
(35)
For the F-KPP equation, this is discussed in more detail in Section 2.5 of [144], where this property is referred to as
“conservation of steepness”.
48
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 4. Illustration of the crossover in the case of an initial condition which falls of exponentially with steepness ¿ ∗ ,
viewed in the frame = x − v∗ t moving with the asymptotic spreading speed. Along the vertical axis we plot the logarithm
of the amplitude of the transient pro3le. The dashed region marks the crossover region between the region where the
linear spreading point contribution dominates and which spreads asymptotically with speed v∗ in the lab frame, and the
exponential tail which moves with a speed venv ¿ v∗ . As indicated, the crossover region moves to the right, so the steep
fast-moving exponential tail disappears from the scene. The speed of the crossover region is obtained by matching the
two regions, and is given by (36).
and hence
vco = 2D( − ∗ ) ± 2D
( − ∗ )2 − [venv () − v∗ ]=D :
(36)
It is easy to check that for equations where !(k) is quadratic in k, the F-KPP equation as well
as the Complex Ginzburg Landau equation discussed in Section 2.11.5, the square root vanishes in
view of relation (24) between D and the curvature of venv () at the minimum. Hence, (36) then
reduces to (32). This is simply because when !(k) is quadratic, the Gaussian integral in the 3rst
argument is actually exact. Since the square root term in (36) is always smaller than the 3rst term
in the expression, we see that the sign of vco , the velocity of the crossover point, is the same
as the sign of − ∗ . Thus, the upshot of the analysis is that the crossover point to a tail with
steepness larger than ∗ moves to the right, and the crossover point to a tail which is less steep,
to the left. 16
The picture that emerges from this analysis is illustrated in Figs. 4 and 5. When ¿ ∗ , i.e. for
initial conditions which are steeper than the asymptotic linear spreading pro3le, to the right for large
enough the pro3le always falls of fast, with the steepness of the initial conditions. However, as
illustrated in Fig. 4 the crossover region between this exponential tail and the region spreading with
16
Note that when the velocity is expanded in the term under the square root sign, the terms of order ( − ∗ )2 always
cancel in view of Eq. (24). Thus the argument of the square root term generally grows as ( − ∗ )3 , and depending on
venv
(∗ ) the roots of Eq. (36) are complex either for ¿ ∗ or for ¡ ∗ . This indicates that the detailed matching in the
regime where the roots of (36) are complex is more complicated than we have assumed in the analysis, but the general
conclusion that the direction of the motion of the crossover point is determined by the sign of − ∗ is unaDected.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
49
Fig. 5. As Fig. 4 but now for the case of an initial condition which falls of exponentially with steepness ¡ ∗ . In this
case, the dashed crossover region moves to the left, so the slowly decaying exponential tail gradually overtakes the region
spreading with velocity v∗ in the lab frame. In other words, the asymptotic rate of propagation for initial conditions which
decay slower than exp(−∗ x) is venv ¿ v∗ .
velocity v∗ moves to the right in the frame moving with v∗ , i.e. moves out of sight! Thus, as time
increases larger and larger portions of the pro3le spread with v∗ . 17
Just the opposite happens when the steepness of the initial conditions is less than ∗ . In this
case vco ¡ 0, so as Fig. 5 shows, in this case the exponential tail expands into the region spreading
with velocity v∗ . In this case, therefore, as time goes on, larger and larger portions of the pro3le
are given by the exponential pro3le (34) which moves with a velocity larger than v∗ .
Because of the importance of initial conditions whose steepness is larger than ∗ , we will
henceforth refer to these as steep initial conditions:
steep initial conditions:
∗
lim (x; 0)e x = 0 :
x→∞
(37)
We will specify the term “localized initial conditions” more precisely when we will discuss the
nonlinear front problem in Section 2.7.6.
In conclusion, in this section we have seen that
According to the linear dynamics, initial conditions whose exponential decay rate
(“steepness”) is larger than ∗ lead to pro6les which asymptotically spread with the linear
spreading velocity v∗ . Initial conditions which are less steep than ∗ evolve into pro6les that
advance with a velocity venv ¿ v∗ .
As we shall see, these simple observations also have strong implications for the nonlinear behavior:
according to the linear dynamics, the fast-moving exponential tail moves out of sight. Thus, with
steep initial conditions we can only get fronts which move faster than v∗ if this exponential tail
17
There is an amusing analogy with crystal growth: the shape of a growing faceted crystal is dominated by the slowing
growing facets, as the fast ones eliminate themselves [420].
50
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
matches up with a nonlinear front, i.e. if there are nonlinear front solutions whose asymptotic spatial
decay rate ¿ ∗ . These will turn out to be the pushed front solutions.
Example. Crossover in the linear F-KPP equation.
The above general analysis can be nicely illustrated by the initial value problem u(x; 0) = (x)e−x
for the linearized F-KPP equation (15), taken from Section 2.5.1 of [144]. Here is the unit step
function. The solution of the linear problem is
√ 1 + erf (x − 2t)= 4t
;
(38)
u(x; t) = exp[ − x − venv ()t]
2
x
2
where erf (x) = 2−1=2 0 e−t is the error function and where venv () is given in (28). The position
of the crossover region is clearly x ≈ 2t, which corresponds to a speed 2( − ∗ ) in the = x − 2t
frame, in accord with (32) and (36) with D = 1, ∗ = 1 and v∗ = 2 [Cf. (17)]. Moreover, this
crossover region separates the two regions where the asymptotic behavior is given by
u(x; t) ≈ exp[ − [x − venv ()t]] ;
= exp[ − [ − (venv () − v∗ )t]]
and
u(x; t) ≈ √
for 2(venv − 2)t ;
(39)
1
exp[ − (x − 2t) − (x − 2t)2 =4t] ;
4t(1 − x=(2t)
1
exp[ − − 2 =4t] for 2(venv − 2)t ;
(40)
4t
in full agreement with the general expressions (34) and√ (33). Finally, note that according
to (38)
√
the width of the crossover region grows diDusively, as t. We expect this width ∼ t behavior of
the crossover region to hold more generally.
≈√
2.4. The linear dynamics: generalization to more complicated types of equations
So far, we have had in the back of our minds the simple case of a partial diDerential equation
whose dispersion relation !(k) is a unique function of k. We now brieKy discuss the generalization
of our results to more general classes of dynamical equations, following [144].
First, consider diDerence equations. The only diDerence with the previous analysis is that in this
case the k-space that we introduce in writing a Fourier transform, is periodic—in the language of a
physicist, the k space can be limited to a 3nite Brillouin zone. Within this zone, k is a continuous
variable and !(k) has the same meaning as before. So, if !(k) has a saddle point in the 3rst
Brillouin zone, this saddle point is given by the same saddle point equations (12) as before, and the
asymptotic expression (14) for the dynamical 3eld is then valid as well! 18
18
When !(k) is periodic in k space, there will generally also be saddle points at the boundary of the Brillouin zone.
These will usually not correspond to the unstable modes—they correspond to a an oscillatory dependence of the dynamical
3eld (like in antiferromagnetism)—but there is no problem in principle with the linear spreading being determined by a
saddle point associated with the edges of the Brillouin zone.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
51
In passing, we note that although the above conclusion is simple but compelling, one may at
3rst sight be surprised by it. For, many coherent solutions like fronts and kinks are susceptible to
“locking” to the underlying lattice when one passes from a partial diDerential equation to a diDerence
equation [154,222]. Mathematically this is because perturbations to solutions which on both sides
approach a stable state are usually governed by a solvability condition. The linear spreading dynamics
into an unstable state, on the other hand, is simply governed by the balance of spreading and growth,
and this is virtually independent of the details of the underlying dynamics.
The concept of linear spreading into an unstable state can be generalized to sets of equations
whose linear dynamics about the unstable state can, after spatial Fourier transformation and temporal
Laplace transform, be written in the form
N
N
Ĥ nm (k; !)˜ m (k; t = 0); n = 1; : : : ; N :
Ŝ nm (k; !)ˆ m (k; !) =
(41)
m=1
m=1
Here n is an index which labels the 3elds. The above formulation is the one appropriate when we
use a temporal Laplace transform,
∞ ∞
ˆ
dt
d x n (x; t)e−ikx+i!t :
(42)
n (k; !) =
0
−∞
In the Laplace transform language, terms on the right-hand side arise from the partial integration of
temporal derivative terms 9kt m (x; t) in the dynamical equation; the coeRcients Hnm therefore have
no poles in the complex ! plane but poles in the k plane can arise from exponentially decaying
initial conditions.
It is important to realize that the class of equations where the linearized dynamics about the
unstable state can be brought to the form (41) is extremely wide: in includes sets of partial differential
diDerence equations, equations with a spatial and temporal kernels of the form
t equations,
dt K(x − x ; t − t )(x ; t ), as well as equations with a mixture of such terms. 19 In addition,
dx
we conjecture that much of the analysis in this section can quite straightforwardly be extended to
front propagation into periodic media (see Section 3.18). We will give a few simple examples based
on extensions of he F-KPP equation below.
The Green’s function Ĝ associated with the equations is the inverse of the matrix Ŝ,
Ĝ(k; !) ≡ Ŝ −1 (k; !)
and the formal solution of (41) can be written simply in terms of Ĝ as
V t = 0) :
ˆ !) = Ĝ(k; !) · Ĥ (k; !) · (k;
(k;
(43)
(44)
When we invert the Fourier-Laplace transform, the term on the right-hand side has, in view of
(43), poles at the points where the determinant |Ŝ| of Ŝ vanishes. There may generally be various
19
I have the impression that population dynamicists [124,295] realized most clearly 3rst that the front speed of what
we refer to as pulled fronts can be calculated explicitly also for equations with a memory kernel. Within the physics
community, this was realized of course from the start by the plasma physicists when they developed the “pinch point
formulation” discussed below. Quite surprisingly, it appears that many of these early developments have never become
standard knowledge in the mathematics literature. The “new method” proposed in [164] is essentially a reinvention of
parts of the work half a century before in plasma physics and Kuid dynamics referenced below, and this paper contains no
references to these earlier developments. Even in this paper, the analysis is presented as a method that applies to partial
diDerential equations only.
52
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
branches of solutions of the equation |Ŝ| = 0. In discussing the large-time behavior, one 3rst assumes
that the initial conditions have compact support, so that their spatial Fourier transform is again an
entire function of k. The analysis then amounts to extracting the long-time behavior of the Green’s
function G.
The poles given by the zeroes of |Ŝ| determine the dispersion relations !" (k) of the various
branches ". The branches on which all modes are damped do not play any signi3cant role for the
long-time asymptotics. For each of the branches on which some of the modes are unstable, the
analysis of the previous sections applies, and for the linear problem the linear spreading velocity v∗
is simply the largest of the linear spreading speeds v"∗ of these branches.
In fact, the long time asymptotics of n (x; t) can be extracted in two ways from (44), depending
on whether one 3rst evaluates the !-integral or the k-integral. The 3rst method essentially reproduces
the formulation of the previous sections, the second one leads to the so-called pinch-point formulation
[49,62,204,264] developed in plasma physics in the 1950s. We discuss their diDerences, as well as
their advantages and disadvantages in Appendix A, and proceed here keeping in mind that the two
methods invariably give the same expressions for the linear spreading velocity v∗ and associated
parameters.
In order to keep our notation simple, we will from now on drop the branch index ", assuming
that the right linear spreading point has been selected if there is more than one, and we will usually
also drop the index n or the vector notation for the dynamical 3eld .
Example. Finite diDerence version of the F-KPP equation.
As a simple example of the implications of the above discussion, imagine we integrate the F-KPP
equation with a cubic nonlinearity with a simple Euler scheme. 20 This amounts to replacing the
F-KPP equation by the following 3nite diDerence approximation:
uj (t + Wt) − uj (t) uj+1 (t) − 2uj (t) + uj−1 (t)
=
+ uj (t) − uj3 (t) :
Wt
(Wx)2
(45)
If we linearize the equation by ignoring the last term and substitute a linear mode uj ∼ exp(st−jWx)
(this amounts to writing ! = is with s real) we obtain the dispersion relation
2
sinh 12 Wx
exp[sWt] − 1
=1+
:
(46)
1
Wt
Wx
2
The saddle point equations or, what amounts to the same, the minimum of the curve venv () = s=
is easy to determine numerically. For small Wt and Wx one can also solve the equation analytically
by expanding about the values for the continuum case given in (17), and one 3nds [144]
 ∗
1
(Wx)2 + · · · ;
v = 2 − 2Wt + 12



(47)
Euler approximation to F-KPP
∗ = 1 + Wt − 18 (Wx)2 + · · · ;



D = 1 − 4Wt + 12 (Wx)2 + · · · ;
20
The equivalent expressions for the second-order implicit (“Crank–Nicholson”) integration scheme are given in
Section 5.6.4 of [144].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
53
2.0
;
v
1.5
1.0
D
;
λ
0.5
0.0
0.0
1.0
2.0
τ
3.0
4.0
5.0
Fig. 6. Plot of v∗ , ∗ and D as a function of $ for the extension (48) of the F-KPP equation with memory kernel (49).
From [144].
Although these expressions look simply like error estimates for the 3nite diDerence approximation
of the F-KPP partial diDerential equation, they are actually more than that: they give the exact
parameters v∗ ; ∗ and D of the 3nite diDerence approximation. So when the precise values of these
parameters are not so important, e.g., if one want to study the emergence of patterns or the power
law relaxation discussed below in Section 2.9, one can take advantage of this by doing numerical
simulations with relatively large values of Wt and Wx using the above properties as the reference
values, rather than those obtained in the continuum limit Wt; Wx → 0.
Example. F-KPP equation with a memory kernel.
The extension of the F-KPP equation in which the linear growth term is replaced by a term with
a memory kernel,
t
2
9t u(x; t) = 9x u +
dt K(t − t )u(x; t ) − uk (x; t) (k ¿ 1) ;
(48)
0
is an example of a dynamical equation which can still be treated along the lines laid out above, as
its Fourier transform is of form (41). If we take for instance [144]
1
−(t − t )2
;
(49)
K(t − t ) = √ exp
4$2
$
the implicit equation for s() = !i () becomes
2 − s + exp[$2 s2 ] erfc($s) ;
(50)
where erfc is the complementary error function. The result for v∗ , ∗ and D obtained by solving
numerically for the minimum of venv = s()=, are shown in Fig. 6.
Note that when $1 we can to a good approximation expand u(x; t ) in the memory term in (48)
around u(x; t) to second order. In this approximation, we then arrive at a second order version of the
54
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Absolutely unstable
(a)
Convectively unstable
(b)
Fig. 7. Illustration of the distinction between an absolute instability and a convective instability, according to the linearized
dynamics. In the 3rst case, sketched on the left in (a), the perturbation about the unstable state grows for suRciently
long times at any position x. In the convectively unstable case sketched on the right in (b), the perturbation grows
but is at the same time advected away so fast that it eventually dies at any 3xed position x. Note that as a result of
it, the point on the right where reaches the level line = C retracts to the left. The transition from convective to
absolute instability occurs when this point does not move for long times, i.e. when the left spreading velocity vL∗ equals
zero [49,62,204,205,264]. Note that the distinction between absolutely and convectively unstable depends on the frame of
reference. The case sketched on the left which is absolutely unstable in the x frame is convectively unstable in a frame
moving with a suRciently large speed to the right, while the case sketched on the right in the x-frame becomes absolutely
unstable in a frame moving with speed larger than v∗ to the left.
F-KPP equation. Such extensions have often been used as a simple way to model delay eDects, as
will be discussed brieKy in Section 3.19. From the point of view of determining the linear spreading
speed, however, there is no real advantage in using a second order equation rather than an equation
with a kernel.
2.5. The linear dynamics: convective versus absolute instability
The case that we will typically have in mind is the one in which the growth rate of the unstable
modes is so strong that the amplitude of a generic localized perturbation grows for long times at
any 3xed position, as sketched in Fig. 7(a). It thus spreads into the unstable state on both Kanks of
the perturbation.
However, even when a state is linearly unstable, so that according to (4) a range of modes has a
positive growth rate !i , if there are symmetry breaking convective terms in the dispersion relation
a localized perturbation may be convected away faster than it grows out. Fig. 7(b) illustrates how
in this case the amplitude of the perturbation for any 3xed position on the right actually decreases
in time, even though the overall amplitude grows. Even for any position on the left of the 3gure, if
we wait suRciently long the amplitude of the perturbation eventually decays. This regime is usually
referred to as the convectively unstable regime, while the other regime is referred to as the absolutely
unstable regime [49,62,204,205,264].
Clearly, these concepts are intimately connected with the linear spreading speed discussed above.
Indeed, given our convention to focus on the right Kank of a perturbation, the two regimes are
distinguished according to whether v∗ is positive or negative:
v∗ ¿ 0 : linearly absolutely unstable regime ;
v∗ ¡ 0 : linearly convectively unstable regime :
(51)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
55
It is important to keep in mind that the two regimes are de3ned in relation to a particular frame
of reference: if v∗ ¿ 0 in the lab frame, so that the instability is absolute in that frame, the instability becomes convectively unstable for an observer moving to the right with velocity larger than
v∗ . Furthermore, a convectively unstable system ideally remains in or will return to the unstable
state according to the linear deterministic dynamics. Nevertheless, small perturbations or Kuctuations
are ampli3ed while they are advected away. This makes convectively unstable systems particularly
sensitive to Kuctuations and to small but 3xed perturbations at a particular point. The latter could
arise in real experiments due to imperfections in experimental setup or due to local perturbations
at an inlet in systems with a throughKow. We will encounter a nice experimental illustration of the
sensitivity to noise near the convective to absolute transition in Section 3.11.
We 3nally note that we have followed here the standard practice to distinguish the two regimes
according to the fully linear dynamics. The generalization of these concepts to the nonlinear regime
will be given later in Section 2.10.
Example. The F-KPP equation with a convective term.
Consider the F-KPP equation with a convective term,
9t u(x; t) + s9x u(x; t) = 92x u(x; t) + f(u) ;
(52)
where, as in the standard form (1), f(0) = 0 and f (0) = 1. Since this equation simply becomes the
standard F-KPP equation (1) upon transforming to the moving frame x = x − st, the linear spreading
∗
velocity associated with (52) is v∗ =s+vFKPP
=s+2. Thus the linear instability changes from absolute
to convective at s = −2. Whether nonlinearly the transition also happens at this point depends on the
function f(u): when f is such that fronts are pulled, the nonlinear transition remains at s = −2, but
when the fronts become pushed, the transition shifts to more negative values of s—see Section 2.10.
In the Taylor–Couette experiments reviewed in Section 3.11 the velocity s in the closely related
amplitude equation is controlled with a Kow through the cell.
2.6. The two-fold way of front propagation into linearly unstable states:
pulled and pushed fronts
In the previous sections, we have analyzed the linear spreading dynamics into a linearly unstable
state. We concluded that starting from suRciently localized initial conditions—the precise condition
being given in (37)—the perturbations spread into the unstable state with asymptotic speed v∗ . This
asymptotic spreading speed and associated parameters are determined through the dispersion relation
!(k) via Eqs. (12).
We now turn to the genuine nonlinear front propagation problem already stated in the introduction,
questions like: If initially a spatially extended system is in a linearly unstable state everywhere
except in some spatially localized region, what will be the large-time dynamical properties and
speed of the nonlinear front which will propagate into the unstable state? Are there classes of
initial conditions for which the front dynamics converges to some unique asymptotic front state?
If so, what characterizes these initial conditions, and what can we say about the asymptotic front
properties and the convergence to them?
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
As before, our discussion is aimed to be as general as is possible. The only restriction we will
make, barring pathological cases, is that the nonlinear dynamical equations of the system under
investigation have a suRcient degree of “locality”: they can involve partial derivative terms and
nonlocal terms with a kernel whose range is essentially 3nite, but the dynamics of the dynamical
variables at a given position should not depend crucially on the nonlinear state of the system in3nitely
far away. 21; 22
A characteristic feature of front propagation into unstable states which carries over from the linear
dynamics that we discussed above, is that initial conditions which decay very slowly spatially lead
to a large front speed whose value depends on the spatial decay rate. We will come back to this
later. Hence, in order to make precise statements we need to limit the class of initial conditions we
consider. Now, for any given set of deterministic dynamical evolution equations with an unstable
state, the linear spreading speed v∗ and associated steepness ∗ are explicitly and uniquely calculable
according to the discussion of the previous sections. This motivates us to distinguish the front
propagation properties of dynamical equations according to the long-time evolution starting from
steep initial conditions:
Whenever we associate a particular front propagation mechanism with a given dynamical
equation, this is a statement about the dynamical evolution of fronts that evolve from localized
initial conditions. These include all initial conditions which are steep”, i.e., which as de6ned
in (37) fall o> faster than exp[ − ∗ x]. 23
Let vfront (t) be some suitably de3ned instantaneous front velocity. The crucial insight on which
our presentation will be based is the following simple insight: For front propagation into a linearly
unstable state, there are only two possibilities if we start from steep initial conditions,
I. vas ≡ limt →∞ vfront (t) = v∗ ⇔ “pulled” front,
II. vas ≡ limt →∞ vfront (t) = v† ¿ v∗ ⇔ “pushed” front.
This statement amounts to the claim that nonlinear fronts propagating into a linearly unstable state
can asymptotically not propagate with speed slower than v∗ for equations of the type we consider, in
which the dynamics is “local” in the sense that the linear dynamics is not suppressed by nonlinear
behavior arbitrarily far away. To see this, suppose we start with a front solution which propagates
with speed less than v∗ . Any suRciently small perturbation ahead of it will then grow out and
spread, asymptotically with the linear spreading speed v∗ . Eventually these perturbations will grow
large enough that nonlinear behavior kicks in, but the crossover region where this happens must
21
This requirement is comparable to the fact that in statistical physics the standard universality classes of equilibrium
phase transitions only apply to Hamiltonians in which the interactions are suRciently short-ranged. This does include
Hamiltonians where the interactions decay as a power r −+ provided + is large enough.
22
A simple example that illustrates this is the following extension of the F-KPP equation: 9t u = 92x u +
∞
u(1 − m tanh[ −∞ d x u2 ]) − u3 . For m ¡ 1 fronts in this equation are still pulled, albeit with a spreading speed
renormalized down from the value obtained by linearizing the equation in u about u = 0. For m ¿ 1 there is no 3nite
asymptotic front speed. This example illustrates that although it is diRcult to specify the general conditions under which
our analysis applies precisely, in practice common sense gets one quite far for any given problem.
23
We will later in Section 2.7.6 enlarge the class of initial conditions which lead to pushed front solutions.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
57
asymptotically advance at least with the speed v∗ . In other words, since the nonlinearities cannot
suppress the linear growth arbitrarily far ahead of the front, they cannot prevent the linear spreading
to asymptotically move ahead with asymptotic speed v∗ and thus create a front with at least this
speed. Consequently, nonlinearities can only drive front speeds up, i.e., to v† ¿ v∗ .
Admittedly, while the above conclusion that fronts can asymptotically not move slower than v∗
may be dynamically “obvious”, I know of no general mathematical proof of it. For some of the
fourth order equations reviewed in Section 2.11, this conclusion is either implicit or explicit in a
formulation in which one proves that in any frame moving with velocity less than v∗ one sees
perturbations grow [93,144].
At this point, the names “pulled” and “pushed” front fall fully into place. 24 With a pulled front
we literally mean one which is pulled along by the linear spreading of small perturbations into
the linearly unstable state. Any front which asymptotically moves with speed v† faster than v∗ is
somehow “pushed” into the unstable state by the nonlinear behavior in the front region itself or the
region behind it—if there were no nonlinear behavior, one would 3nd spreading with velocity v∗ .
We will thus refer to v† as the pushed front speed.
The present line of argument, where one takes the classical linear spreading of perturbations into
an unstable state as the starting point, has several important advantages and rami3cations. First of
all, it allows one to make statements irrespective of the nature of the nonlinear state behind the
front: Quite literally the picture that underlies it and that both the analytical and numerical results
presented later con3rm, is that a pulled front spreads in the way the linear spreading point conditions
(12) force it to do. In other words, the nonlinear dynamics in the region behind the leading edge
just has to adapt to whatever is forced by the linear spreading. Depending on the existence, stability
and nature of nonlinear front solutions, this behavior can be coherent or incoherent, but this by itself
does not really feed back onto the linear spreading. 25 This simple idea can be made explicit and
quantitative: it lies at the basis of the exact results for the universal relaxation behavior of pulled
fronts discussed in Sections 2.9 and 4.
The second advantage is that the present line of argument focuses on the fact that the most clear
and relevant issue is to understand the mechanism through which we can get fronts to propagate at
speeds larger than v∗ , i.e., to be pushed.
We 3nally note that the concept of a pulled front is only well-de3ned for propagation into a
linearly unstable state: A small perturbation around a linearly stable state dies out and does not
spread. As a result, the propagation of a front into a linearly stable state is always driven by the
nonlinearities and the nonlinear or dynamical competition between this state and a diDerent one.
From this perspective, especially near subcritical bifurcation points, it is sometimes useful to think
of a front which propagates into a stable state as being “pushed”.
24
The names pulled and pushed were introduced back in 1976 by Stokes [384] and revived in the physics literature in
the mid-90s by Paquette et al. [333,334]. Ben-Jacob et al. [38] referred to them as Case I and Case II marginal stability
and I initially used the words linear and nonlinear marginal stability [421]. The great advantage of the nomenclature of
Stokes is that the notions of “pulled” and “pushed” tie in nicely with the general concept of linear spreading velocity, and
hence that they can be de3ned independently of whether or not a uniformly translating or coherent front solution exists.
25
Actually, we can make this quite precise: As we shall see, the 3rst consequence of the feedback of the nonlinear
behavior onto front propagation is the change of the prefactor of the 1=t relaxation term by a factor 3. See Section 2.9.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.7. Front selection for uniformly translating fronts and coherent and incoherent
pattern forming fronts
In order to discuss the selection of a particular type of front and its propagation mechanism, it is
useful to distinguish three diDerent classes of nonlinear front dynamics: 26
• Uniformly translating fronts are nonlinear front solutions for the dynamical variable of form
(x; t) = ,v (-) ;
(53)
where - is the co-moving coordinate
- ≡ x − vt ;
(54)
which we will use for general velocity v, to distinguish it from the coordinate = x − v∗ t moving with velocity v∗ . We remind the reader that, unless noted otherwise, we do not distinguish
notationally between a single dynamical variable or a set (vector) of them. Eq. (53) expresses
that in a frame moving with velocity v, the front solution is stationary, i.e., invariant in time.
Uniformly translating solutions are normally only appropriate solutions for problems in which the
front leaves behind a homogeneous state, i.e., does not generate a pattern. The overwhelming
majority of asymptotic front solutions which have been analyzed in the mathematics literature are
of this type. An example of a pushed front in the F-KPP equation is shown in the lower left
panel of Fig. 1.
• Coherent pattern forming fronts are the generalization of uniformly translating solutions to systems where the front leaves behind a pattern with a well-de3ned wavenumber. As the middle panel
of Fig. 1 and the examples discussed in Sections 2.11 and 3 illustrate, usually the pattern behind
the front is stationary or moves with a velocity diDerent from the front speed. In other words,
when viewed in the frame moving with the front, the pattern behind the front is not stationary.
Hence the front solution cannot be of form (53). However, we may generalize the concept by
introducing a coherent pattern forming front solution as a front solution which is periodic in the
frame - = x − vt moving with speed v, i.e.,
(x; t) = ,(-; t)
with ,(-; t + T ) = ,(-; t) :
We can equivalently write this as
(x; t) =
e−in/t ,vn (-)
(/ = 2=T ) ;
(55)
(56)
n=0;±1;:::
26
With our classi3cation we have in mind fronts which propagate in a homogeneous background medium. If the medium
itself is periodic, fronts which are uniformly translating in a homogeneous medium become automatically periodic; likewise
if the parameters of the dynamical equation are randomly varying the fronts will always be incoherent. Strictly speaking,
fronts in a periodic medium, which are sometimes referred to as “periodically pulsating fronts”, are of the type (55) below
but as stated, we will always have in mind that the medium itself is homogeneous, and when considering periodic front
solutions concentrate on fronts which then generate a nontrivial pattern. For the linear dynamics of small perturbations about
the unstable state of a homogeneous system, the dispersion relation can be obtained from a simple Fourier transformation.
As we shall discuss below in Section 3.18, the analysis of pulled fronts can be extended to the case of periodic media
by recognizing that one simply has to do a Floquet analysis to determine !(k).
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
59
where for real dynamical 3elds the complex functions ,vn obey the symmetry
,v−n (-) = ,vn (-) :
(57)
Clearly, Eq. (55) expresses that in the co-moving frame the front solution is periodic. 27 Hence, in
representation (56) the functions ,vn (-) can at every position be viewed as the Fourier coeRcients
of the time-periodic function ,(-; t).
These time-periodic coherent front solutions of form (55) or (56) are relevant for e.g. the Swift–
Hohenberg equation discussed in Section 2.11.2 and were to my knowledge for the 3rst time
introduced for the analysis of fronts in this equation by Collet and Eckmann [92]. Moreover, they
can also be viewed as an extension of the “coherent structure solutions” [424] introduced in the
context of the Complex Ginzburg Landau equations of Sections 2:11:5 and 2:11:6—because of the
fact that the Complex Ginzburg Landau equation arises as the lowest order amplitude equation
for traveling wave patterns, only one term in the sum (56) is nonzero in this case. As we shall
see, both pulled and pushed coherent fronts solutions are found in certain parameter ranges.
• Incoherent pattern forming fronts is the name we will use for all the incoherent fronts like
those of the right column of Fig. 1 which do not 3t into one of the two previous classes.
We will encounter other pulled and pushed examples of them later, but they are the least
understood of all.
We now proceed to discuss the dynamical mechanism that distinguishes pushed and pulled fronts
for these various classes of equations. The essential ingredient will be to maximally exploit the
constraints imposed by the linear dynamics about the unstable state.
2.7.1. Uniformly translating front solutions
When a dynamical equation also admits a homogeneous stable stationary state ss = const: in
addition to the unstable state = 0, then usually this dynamical equation also admits uniformly
translating front solutions of form (53), (x; t) = ,v (-) with - = x − vt. To investigate whether it
does, one substitutes this Ansatz into the dynamical equation and analyzes its behavior. The simplicity
of uniformly translating fronts lies in the fact that the function ,v (-) depend on the single variable
- only, so that the function ,v then obeys an ordinary diDerential equation rather than a partial
diDerential equation. The existence of front solutions can from there on be analyzed using standard
methods [13,14,191]: the homogeneous stationary states = 0 and = ss are 3xed points of this
ordinary diDerential equation. By linearizing about these 3xed points and studying the dimensions of
the stable and unstable manifolds (i.e. the dimensions of the manifolds Kowing into and out of each
3xed point), one can then study the multiplicity of front solutions (see e.g. [424] for an extensive
use of such “counting arguments” in the context of CGL equations). For the F-KPP equation the
analysis, which is summarized below, is relatively trivial; one 3nds that there is a one-parameter
family of front solutions that connect the stable homogeneous state with the unstable state = 0. For
the F-KPP equation, one can easily go beyond a simple counting argument based on the analysis
27
We prefer to use the general name “coherent pattern forming front solution” instead of periodic front solutions for two
reasons. First of all, this name is consistent with the name “coherent structure solutions” for special type of solutions of
the CGL equation (see Eq. (101) in Section 2.11.6). Secondly, we wish to stress the fact that we aim at pattern forming
solutions and so want to distinguish them from the simpler periodic or “pulsating” types of fronts that arise in equations
like the F-KPP equation when the medium itself is periodic—see Section 3.18.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
near the 3xed points to prove more rigorously when there is such a family of solutions. These
front solutions can be parametrized by their velocity v. We stress that for an arbitrary dynamical
equation that admits a stable homogeneous state and a homogeneous unstable state, there is not
necessarily always a one-parameter family of front solutions connecting the two. However, this is
what is typically found and what is intuitively most reasonable based on the fact that a special
feature of the linear dynamics about an unstable state is that any exponential tail with steepness less than ∗ can propagate into the unstable state with speed venv ()—see Section 2.2. 28
Let us 3rst proceed by assuming the equations do admit a one-parameter family of front solutions,
parametrized by their velocity. Then this family generically will include a uniformly translating front
solution ,v∗ () moving with velocity v∗ , to which the front will asymptotically converge in the pulled
regime, i.e., starting from steep initial conditions.
The behavior near the 3xed points of the ordinary diDerential equation for ,v (-), obtained by
linearizing the Kow equations around the 3xed point, corresponds to an exponential --dependence.
Thus, for an arbitrary velocity v, the solutions ,v (-) will decay to zero exponentially for large -,
,v (-) ≈ a1 e−1 - + a2 e−2 - + · · ·
(- → ∞) ;
(58)
where all roots i are positive and where the real coeRcients a1 , a2 , etc. can only be determined
by solving the equations for the fully nonlinear front solution. 29 We take these roots ordered,
0 ¡ 1 ¡ 2 · · ·. The relation between the velocity v and the two smallest roots in the generic case 30 is
shown in Fig. 3: For v ¿ v∗ the root 1 ¡ ∗ while 2 ¿ ∗ . In passing we note that at v=v∗ two roots
coincide; general results [13,14,191] for the Kow behavior in the presence of degenerate eigenvalues
then imply
,v∗ () ≈ (a1 + a2 )e− + a3 e−3 + · · · :
∗
(59)
When do we expect the equation to be pushed, i.e., when do we expect that a pushed front solutions
with velocity v† ¿ v∗ will emerge starting from steep initial conditions? The answer lies hidden in
Fig. 4: according to the fully linear dynamics an exponential tail which is steeper than ∗ and which
corresponds to the dashed branch 2 in Fig. 3, does run faster than v∗ but disappears from the
scene because the crossover point to the linear spreading pro3le moves to the right. According to
28
An explicit counting argument for partial diDerential equations that are symmetric under spatial reKection con3rms
that for such equations one typically expects a one-parameter family of front solutions (see Appendix A of [420]). There
are two intimately related ways to understand why this is so often the case: (i) Suppose we 3rst change the dynamical
equation so as to make the unstable state stable. One then expects there to be at least one front solution (or maybe a
discrete set of them) connecting the two stable states. Now when we change the equation back to its original form, at
the moment when the = 0 state becomes unstable again, the dimension of the stable manifold Kowing into the = 0
3xed point increases by one, because a new root appears which is related to the left branch in Fig. 3; (ii) According to
the linear dynamics discussed in Section 2.2, any exponential tail with ¡ ∗ can propagate into the unstable state with
envelope velocity venv (). When the dynamical equation admits a stable homogeneous state as well, it is not unreasonable
that every exponential tail propagating with some velocity v in the leading edge can match up smoothly with a saturating
behavior behind the front.
29
In (58) we want to bring out that there are contributions from the various roots, or, in more technical language, from
the various directions that span the stable many-fold of the 3xed point corresponding to =. If 1 ¡ 2 =2 then of course
terms exp[ − 21 -] dominate over terms exp[ − 2 -], but this is all hidden in the dots.
30
We assume here for simplicity that there are no other branches at even smaller ; this is usually the case, as the
branch which in Fig. 3 diverges as ↓ 0 is associated with the instability, the fact that !i ( = 0) ¿ 0. We will brieKy
come back to the case where other branches intervene in Section 2.7.7.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
61
(b)
Fig. 8. Illustration of the fact that when a nonlinear front solution ,v† exists, whose asymptotic steepness is according to
(60) larger than ∗ , then this front will generically emerge: it is the selected pushed front solution. Figure (a) should be
compared to Fig. 4: If the equation would be fully linear, the steep tail on the right, which moves at velocity larger than
v∗ , would cross over in the dashed region to the dashed pro3le moving with asymptotic speed v∗ . However, this does
not happen. When the steep pro3le matches up arbitrarily well with a fully nonlinear front pro3le ,v† this steep region
does not disappear from the scene: Instead a fully nonlinear pro3le with speed v† emerges. While (a) shows how only
the pushed front solution ,v† can asymptotically emerge, (b) illustrates how a pushed front solution invades a region
where the pro3le is close to that given by the linear spreading analysis. The dashed line indicates the continuation of the
pro3le as given by the linear spreading analysis of previous sections, but in the dashed region the pro3le crosses over
to the steep tail of the nonlinear pushed pro3le. The fact that the crossover region moves to the right with the speed vco
determined earlier, con3rms that the pushed front solution invades the leading edge.
our “locality” assumption for the dynamics, the same holds arbitrarily far into the leading edge of a
front, unless the nonlinearities in the dynamical equation allow the fast exponential tail to match
up perfectly with a nonlinear front solution! This is illustrated in Fig. 8: if a uniformly translating
front solution with v = v† ¿ v∗ exists for which a1 = 0 so that its asymptotic behavior is given by
the root 2 ,
,v† (-) ≈ a2 e−2 - + a3 e−3 - + · · ·
(2 ¿ ∗ ; - = x − v† t → ∞) ;
(60)
then this front solution can and will overtake any transient dynamical tail in the leading edge. In
other words, such a solution is the asymptotic pushed front solution sought for, and if it exists it is
the dynamically relevant front solution emerging from steep initial conditions.
As we noted before, front solutions ,v (-) obey an ordinary diDerential equation or a set thereof;
such a diDerential equations can be formulated as a Kow in phase space [13,14,191]. In such an
interpretation, a front solution corresponds to a so-called heteroclinic orbit, an orbit which goes
from one point (the asymptotic state behind the front) to another (the state = 0). For general
v, the orbit approaches the 3xed point corresponding to the state = 0 asymptotically along the
eigendirection whose rate of attraction, given by 1 , is slowest. The pushed front solution ,v† (-),
however, approaches it along the slowest but one eigendirection, the one with eigenvalue 2 . For
this reason, a uniformly translating pushed front solution is sometimes referred to as a strongly
heteroclinic orbit [354].
Let us illustrate the above considerations brieKy for two equations, the F-KPP equation and an
extension of it, the EFK equation. Uniformly translating solutions U (x − vt) = U (-) of the F-KPP
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 9. Sketch of the Kow in the phase space (U; U ) of the Kow equations (62) and (63) that govern the uniformly
translating solutions of the F-KPP equation. The solid lines at the two 3xed points indicate the shape of the stable and
unstable manifolds near each 3xed point, obtained by linearizing the Kow around each 3xed point. The arrows indicate the
direction of the Kow, and the double arrow at the (0; 0) 3xed point indicates that the contraction along this eigendirection is
stronger than along the other one (2 ¿ 1 ). The dashed trajectory is the heteroclinic orbit connecting the two 3xed points;
it corresponds to the front solution connecting the stable and the unstable state. (a) Flow in the case v ¿ max(v† ; v∗ ).
The trajectory approaches the (0; 0) 3xed point along the slowest contracting direction. (b) If for some velocity v = v† the
dashed trajectory becomes a “strongly heteroclinic orbit” which approaches the (0; 0) 3xed point along the more strongly
contracting direction, the equation admits a pushed front solution.
equation (1) obey the ordinary diDerential equation
−v
dU
d2 U
+ f(U ) :
=
dd-2
(61)
It is convenient to use the standard trick of writing this second order equation as a set of 3rst order
equation: by introducing the variable U = dU=d- we can write (61) as
dU
= U ;
(62)
ddU = −vU − f(U ) :
(63)
dThese equations describe the Kow in the two-dimensional phase space (U; U ), with - playing the
role of time. Because f(0) = f(1) = 0 according to (1), the points (0; 0) and (1; 0) are 3xed points
of these Kow equations; the 3rst one corresponds to the stable state and the second one to the
unstable state. With our convention that fronts move into the unstable state on the right, a uniformly
translating front solution corresponds to a trajectory Kowing from the (1; 0) 3xed point to the (0; 0)
3xed point. Such a “heteroclinic orbit” is sketched for large arbitrary v in Fig. 9(a). 31 The solid lines
near the two 3xed points in this 3gure denote the directions of the stable and unstable manifolds
Kowing into and out of each 3xed point. These are easily determined by linearizing equations (62),
(63) about the 3xed point solutions and solving for the eigenvalues of the linearized Kow. The
arrows in Fig. 9 indicate the directions of the Kow for increasing - (“time”). As is indicated in
the 3gure, there is one stable and one unstable stable direction at the (1; 0) 3xed point. For any
3xed velocity v, there is hence a unique trajectory coming out of this 3xed point in the direction of
decreasing u. At the (0; 0) 3xed point, however, both eigendirections are attracting; we have indicated
the direction along which the contraction is largest with a double arrow. Now, because there is
31
For v ¡ v∗ the eigenvalues at the 3xed point (0; 0) are complex so trajectories spiral into this 3xed point.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
63
a two-dimensional manifold Kowing into the (0; 0) 3xed point, the unique dashed trajectory that Kows
out of the (1; 0) 3xed point will Kow into the (0; 0) 3xed point, and it will asymptotically Kow in
along the slowest contracting eigendirection—these observations correspond to the statements that
there is a front solution for a any v and that the asymptotic large-- behavior in (58) is dominated
by the smallest eigenvalue 1 .
Depending on the form of the nonlinearity f(u), the situation as sketched in Fig. 9 may occur:
for a particular value v† of the velocity, it may happen that the unique dashed trajectory that Kows
out of the (1; 0) 3xed point Kows into the stable (0; 0) 3xed point along the most rapidly contracting
direction which is indicated in the 3gure with the double arrow. In other words, for this trajectory the
asymptotic behavior of U (-) goes as e−2 - , i.e., has a1 = 0 in (58). This “strongly heteroclinic orbit”
thus corresponds to a pushed front solution—if it exists, the selected front in the F-KPP equation is
a pushed front. Clearly, whether such a pushed front solution exists depends on the global nonlinear
properties of the equation, the full nonlinear behavior of f(u) in this case—determining or proving
whether for a given equation the selected front is pushed, requires a fully nonlinear global analysis
of the Kow. All the details of the nonlinear behavior count. In the example at the end of this section
we will show that for certain classes of nonlinearities f(u), the pushed front solution of the F-KPP
equation can be obtained analytically.
To give an idea of the complications that one immediately encounters when one goes beyond the
F-KPP equation, let us brieKy consider the equation
9t u = 92x u − 294x u + f(u) ;
(64)
which can be thought of as an extension of the F-KPP equation. Indeed, with f(u) = u − u3 this
equation is the EFK (“Extended Fisher–Kolmogorov”) equation which we will discuss in more detail
in Section 2.11.1. As we shall see there, the pulled fronts in this equation exhibit a transition from
uniformly translating fronts to coherent pattern forming fronts at 2 = 1=12, but we focus for now on
the uniformly translating fronts for 2 ¡ 1=12. If we substitute the Ansatz =U (-) into the equation
and write the resulting ordinary diDerential equation as a set of 3rst order equations, we get in
analogy with (62) and (63)
dU
= U ;
d-
(65)
dU = U ;
d-
(66)
dU = U ;
d-
(67)
dU = 2−1 [U + vU + f(U )] :
d-
(68)
Since the Kow is now in a four-dimensional phase space, it is clear that the question of existence
of such uniformly translating front solutions is much more subtle than for the F-KPP equation: all the
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
uniformly translating pushed front
in the EFK equation
(a)
(b)
Fig. 10. (a) Space time of a numerical solution of Eq. (64) with nonlinearity f(u) = u + 9u2 − 10u3 . This is an extension
of the EFK equation discussed in Section 2.11.1. For this nonlinearity, the selected front is a pushed front propagating
with speed v† = 2:751 [421]. The time increment between successive time-slices is 0.75; the initial condition is a small
Gaussian peak centered at the origin, and the total simulation time is 22.5. The dashed line indicates a position moving
with the speed v∗ = 1:8934. Although the front might appear to be monotonically decaying to the right, close inspection
of this pushed front in the tip shows that it is actually non-monotonic; the enlargement in panel (b) shows this more
clearly. (b) Blow up of the asymptotic front pro3le in the leading edge. The symbols denote the actual front values at
the grid points in the simulations, while the full line is the pro3le in the leading edge given by the three terms in (60).
The dashed line denotes the mode exp(−1 -). Clearly, for the pushed front solution, a1 =0 indeed, in accordance with the
mechanism of pushed front propagation set forth in the main text. From [421].
simpli3cations of Kow in a two-dimensional plane, special to the F-KPP equation, are lost. 32 Let
us here simply illustrate that one important simplifying property of the F-KPP equation is immediately lost. From the phase space arguments sketched above for the F-KPP equation, it immediately
follows that the selected front is the front solution with the smallest speed for which the front
solution is monotonically decaying with -. Therefore, this idea has sometimes been proposed in
the literature as a general principle for front selection. However, it is simply a property of the
F-KPP equation and a limited class of extensions of it—it is a property that does not really have
anything to do with front selection and just does not hold generally. The simulations of Fig. 10
illustrate this for the above Eq. (64) with f(u) = u + 9u2 − 10u3 . As the left panel illustrates,
the selected front is a uniformly translating pushed front in this case. Although this is hardly
visible in the space-time plot on the left, this pushed front solution is not monotonically decaying towards the right. The enlargement of the leading edge in the right panel shows how U 3rst
goes through zero and then has a local maximum at negative U . This plot also con3rms fully the
mechanism for pushed fronts that we have identi3ed above: the selected pro3le 3ts the form (58)
with a1 = 0.
32
If one again linearizes the Kow near the two 3xed points (0; 0; 0; 0) and (1; 0; 0; 0), the multiplicity of the stable and
unstable manifolds is such that one would indeed expect the existence of a one-parameter family of front solutions [421].
A more rigorous study of such front solutions and of the Kow in this four-dimensional phase space has been taken up
only recently in the mathematics literature. See in particular [371] for a proof of the existence of front solutions of the
above Kow equations and [340,410] for other types of solutions.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
65
Let us return to our general formulation of the mechanism through which pushed fronts arise for
uniformly translating fronts. While this scenario is arrived at simply by exploiting our insight into
the leading edge where the dynamics is essentially given by the linear dynamics, it is in complete
agreement with (i) what is known rigorously for the second order F-KPP equation, (ii) the (marginal)
stability arguments for front selection (Section 2.8) which show that if a strongly heteroclinic front
solution ,v† exists, all front solutions with lower speed are unstable to invasion by this front, and
(iii) all numerical results known to me. Furthermore, the extension of the argument to coherent
pattern forming fronts is fully consistent with the analytical and numerical results for the quintic
CGL equation discussed in Section 2.11.6).
We 3nally return to the question what happens if the front solutions do not admit a one-parameter
family of front solutions. If this happens, then we generically expect that the equations will not
admit a uniformly translating pulled front solution moving with velocity v∗ . A pulled front can
then not be uniformly translating—the pulled front solution must then show nontrivial dynamics in
the front region. Presumably, the dynamics is then either that of a coherent or incoherent pattern
forming front, even if the state it 3nally leaves behind is structureless. Furthermore, the absence of
a one-parameter family of uniformly translating front solutions makes it also very unlikely that there
will be uniformly translating pushed front solutions, as these have to obey one additional constraint
a1 = 0.
Example. Pushed front solutions of the F-KPP equation and the reduction of order method.
For the F-KPP equation, for which the dispersion relation !(k) is quadratic, a general uniformly
translating front solution u(x − vt) = U (-) falls oD to u = 0 with two exponentials—in other words,
for a front solution with arbitrary velocity v there are two terms in expression (58) for the large behavior. However, as we already discussed above, a pushed front solution of the F-KPP equation
approaches u = 0 with a single exponent, as a1 = 0. A good strategy to look for exact pushed front
solutions of the F-KPP equation is therefore [421] to investigate when front solutions of the 3rst
order equation
dUv
= h(Uv )
d-
(69)
are also front solutions of the full ordinary diDerential equation for front solutions Uv (-) of the
F-KPP equation. This is the case if they satisfy Eq. (61),
−v
dUv (-) d 2 Uv (-)
=
+ f(Uv (-)) :
dd-2
(70)
Searching for solutions of the type (69) is very easy; to see this, note that upon substitution of (69)
into (70) we get
− vh(Uv ) = h (Uv ) h(Uv ) + f(Uv ) ;
(71)
so that we just need to look for combinations of functions h and f which obey this last equation
(for any given h one can trivially 3nd a function f but the converse is not true). This is especially
simple for polynomial functions. Indeed, it is straightforward to check that the functions [144]
h(u) = −u(1 − un );
f(u) = u
˜ + un+1 − (1 + )u
˜ 2n+1
(72)
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
solve (71) provided we take
˜ + 1
; v = (n + 2) − 1= :
(73)
2 =
n+1
In order that the front solution of these equations corresponds to a pushed front
√ solution of the
F-KPP equation, the decay rate given by (73) needs to be bigger than ∗ = ;
˜ this is easily
found to be the case for ˜ ¡ 1=n. Thus, the F-KPP equation with a nonlinearity f(u) given by 33
(72) has the pushed to pulled transition at ˜ = 1=n, and for ˜ ¡ 1=n the pushed front velocity v† is
given by (73).
For further discussion of these results for the most common cases n = 1 (f a cubic polynomial
in u) and n = 2 (f a function of u, u3 and u5 ), we refer to [38,421]. Other examples are discussed in
[34,144,198,212,326,355,436,447] and Section 3.15, where we will discuss a nontrivial case relevant
to liquid crystals. Quite surprisingly this “reduction of order method”, as it is sometimes called,
also allows one to 3nd the pushed front solutions of the quintic CGL equation (see Section 2.11.6)
and to construct other types of exact solutions of the CGL equation [95,312,322,424]. Deep down,
the method is related to the construction of exact solutions of nonlinear equations using PainlevIe
analysis [95,96,360]. Also symmetry reduction methods have been used to search for exact solutions
of the nonlinear diDusion equation [89,325].
2.7.2. Coherent pattern forming front solutions
Let us now turn to coherent pattern forming front solutions of form (56). Every function ,vn (-)
in this expression also depends only on the co-moving coordinate -. Hence if we substitute the
expansion (56) into the dynamical equation under investigation, the functions ,vn (-) will in general
obey a set of coupled ordinary diDerential equations. In general, this in3nite set of ordinary diDerential
equations is quite cumbersome; however, for fronts we need to have ,vn (-) → 0 as - → ∞, and
to linear order all the ,vm obey essentially the same equation obtained by linearizing the dynamical
equation about the unstable state. We then have the freedom to take the term with n=1 to correspond
to asymptotic decay exp[ − 1 -] associated with the smallest root 1 ; in analogy with (58) we then
have
,v1 (-) ≈ A1 eikr; 1 -−1 - + A2 eikr2 -−2 - + · · · ;
(74)
where now the coeRcients A1 , A2 , etc. are complex. Again it follows from the general arguments
underlying Fig. 3 that for any v ¿ v∗ , we generally have 1 ¡ ∗ , 2 ¿ ∗ .
One may, like in the case of uniformly translating fronts, wonder about the multiplicity of coherent
pattern forming fronts. There are even fewer results in this case, but indications are that for front
propagation into unstable states the typical case is that there is a two-parameter family of front
solutions. That this is the case has been proved for the Swift–Hohenberg equation [92] and is also
found [424] for the quintic CGL equation of Section 2.11.6. Moreover, counting-type arguments for
the pattern-forming regime of the EFK equation (Section 2.11.1) are consistent with this—we expect
that the methods developed by Sandstede and Scheel [375] will allow one to establish this more
generally and cleanly. Intuitively, the existence of a two-parameter family of front solutions is the
33
The prefactor of the term un+1 looks special, but the analysis covers all cases with f(u) of form (72) since all other
cases can be brought to this form by a proper scaling of space, time and u—see [355] and Appendix C of [144].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
67
natural extension of the existence of a one-parameter family of uniformly translating front solutions,
since the leading edge of pattern forming fronts generically is characterized by a wavenumber kr in
addition to the steepness . In other words, our conjecture that the generic situation is that there is
a two-parameter family of coherent pattern forming fronts, means that for every velocity v, there is
a one-parameter family of fronts parametrized e.g. by the wavenumber kr in the leading edge or by
the wavenumber of the coherent pattern behind the front. The front solutions whose wavenumber
leads to maximal growth in the leading edge for given steepness (9!i =9kr = 0), then correspond to
the left branch in Fig. 3.
Let us proceed 3rst by assuming that indeed the equations for the coherent pattern forming front
solutions admit a two-parameter family of front solutions. The above considerations then imply that
there will in general exist a coherent pulled front solution, i.e., a solution whose asymptotic behavior
to the right matches the linear spreading point behavior,
,v1∗ () ∼ (A1 + A2 )e−
∗
+i(kr∗ −(!r∗ −kr v)t)
( = x − v∗ t → ∞) ;
(75)
where in analogy with (59) the term linear in arises because two roots coincide at v∗ . The coherent
pulled front solutions determined this way will be the ones that one will observe in the regime of
pulled front propagation.
The mechanism to get coherent pushed solutions moving at a speed v† ¿ v∗ is completely analogous to the one we discussed above for uniformly translating solutions with the aid of Fig. 8:
starting from steep initial conditions an arbitrary front solution with speed v ¿ v∗ cannot emerge,
since that would be incompatible with the dynamics in the leading edge, but any special coherent
front solution whose asymptotic spatial decay is steeper in that
,v1† (-) ∼ A2 e−2 -+ikr2 - + · · · ;
(76)
can and will invade the leading edge. In other words, if a solution for which A1 = 0 in (74)
exists, this is the pushed front solution that will be selected by the dynamics. Note that since the
coeRcients A1 , A2 , etc. in (74) are complex coeRcients, the condition that A1 = 0 amounts to two
conditions. Whether there are front solutions which obey this condition depends on the equation
under investigation, but since we assume that there is a two-parameter family of front solutions, if
they exist, pushed front solutions are expected to be isolated solutions.
This selection mechanism for coherent pattern forming front solutions is clearly quite analogous to
the one we discussed for uniformly translating fronts, but for coherent fronts I am not aware of any
rigorous work on the pulled to pushed transition of coherent fronts. The numerical and analytical
work on a modi3ed version of the Swift–Hohenberg equation and on the quintic CGL equation,
summarized brieKy in Sections 2.11.2 and 2.11.6 are in full accord with the above scenario.
It is important to keep in mind that our discussion has been based on the idea that coherent
front solutions come as a two-parameter family if they exist. For a particular equation, it is not
guaranteed that they do exist, of course. Indeed, the quintic CGL equation discussed in Section
2.11.6 illustrates this: In some parameter ranges one can show that the equation does not admit any
coherent pulled front solution. In the parameter range where this happens, the pulled fronts become
incoherent—see Fig. 17. Likewise, if the coherent front solutions do not come as a two-parameter
family, then neither coherent pulled front solutions nor coherent pushed front solutions are expected
to exist: the dynamics is then expected to be incoherent.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
We 3nally note an important point. We already noted above that the condition A1 = 0 is equivalent
to two conditions, and that this implies that pushed fronts solutions, if they exist, generically come as
isolated (discrete) solutions. Moreover, this simple counting argument also shows that the dominant
spatial decay rate 2 of this isolated solution (76) generally does not lie on the thick dashed branch
in Fig. 3. 34 By the same token, this implies that at the pulled to pushed transition, i.e. when
v† ↓ v∗ , 2 = ∗ in (76). So, while for uniformly translating pro3les the pulled to pushed transition
corresponds to a continuous transition in the front shape, for coherent pattern forming fronts the
transition is discontinuous in the front shape! Such discontinuous behavior, which was 3rst discovered
for the quintic CGL equation of Section 2.11.6, imply that “structural stability” conjectures do not
apply to pattern forming fronts. We will come back to this in Section 2.8.3.
Examples of pushed coherent fronts will be encountered later in Sections 2.11.1, 2.11.2, 2.11.6,
3.8, and 3.17; we also conjecture that pushed coherent fronts can arise in the models of Sections
3.2 and 3.3, for which so far only pulled fronts have been found.
2.7.3. Incoherent pattern forming front solutions
Incoherent pattern forming fronts are those fronts which do not leave behind a coherent pattern.
Hence they cannot be of form (56). From the numerical simulations of the cubic and quintic CGL
equation discussed in Sections 2.11.5 and 2.11.6 and the full-blown numerical simulations of turbulence fronts discussed in Section 3.9 we know that both pulled and pushed incoherent fronts can exist.
Of course, as always we can calculate the speed of incoherent pulled fronts, but to my knowledge
there is essentially no good understanding of what drives an incoherent front to be pushed. Thus I
cannot give a precise mathematical formulation, analogous to (60) or (76), that identi3es what property will give rise to a pushed incoherent front. My conjecture is that in analogy with what we found
for uniformly translating and coherent pattern forming fronts, incoherent pulled fronts are solutions
which in some average sense do fall oD with a well-de3ned steepness larger than ∗ . The simulations
of turbulence fronts discussed in Section 3.9 con3rm this idea, but I do not know how to give predictive power to this statement. Unfortunately, the problem of the transition from pulled to pushed
incoherent fronts appears to be as hard as the spatio-temporal chaos and turbulence problem itself!
Examples of pushed incoherent fronts will be encountered in Sections 2.11.4, 2.11.6, 3.9 and 3.22.
2.7.4. E>ects of the stability of the state generated by the front
In our discussion of how pushed fronts can arise in the various modes of front propagation,
a central role is played by the existence of the special type of uniformly translating or coherent
pattern forming solutions which for - → ∞ decay with steepness larger than ∗ —these become the
pushed front solutions. To what extent does the stability of fronts play a role? As we will discuss in
Section 2.8 below, one can relate the above selection mechanism of pushed fronts to the stability
properties of front solutions with v ¡ v† , but it is more appropriate to think of this as a consequence
rather than a cause of front selection.
The stability or instability of the state generated by the front does have important consequences
for the dynamics, however. The examples discussed in Section 2.11 will illustrate this most clearly:
it is quite possible that the state which emerges behind the front is itself unstable. When the selected
34
In [421], we suggested that the combinations v† ; 2 of pushed fronts of the Swift–Hohenberg equation should lie on
the dashed branch. This suggestion is wrong.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
69
state is nonlinearly convectively unstable in the frame moving with the front, the state behind the
3rst front is invaded by a second front which moves slower than the 3rst. If, however, the state
generated by the 3rst front is absolutely unstable in the frame moving with the 3rst front, then the
second one will catch up with the 3rst and alter its properties. In the examples we will discuss the
second front is usually an incoherent front, and when the one with which it catches up is a coherent
front, this induces a transition from coherent to incoherent front dynamics in the leading front. In
practice, therefore, what type of front dynamics one will get is determined both by the existence of
front solutions and by the stability of the state these solutions generate.
2.7.5. When to expect pushed fronts?
The requirement for existence of a pushed front solution, a1 = 0 according to Eq. (60) for a
uniformly translating front solution to exist, or A1 = 0 according to Eq. (76) for a coherent pattern
forming solution, is a condition on the global properties of the solution of a nonlinear ordinary
diDerential equation. Hence whether a pushed front solution exists depends on the full nonlinear
properties of the dynamical equation we wish to investigate. There does not appear to be a general
mathematical framework that allows us to predict from the appearance or the global structure of
an equation whether a strongly heteroclinic front solution does exist, and hence whether fronts will
be pulled or pushed. For the F-KPP equation one can derive general conditions on the nonlinear
function f(u) such that the selected fronts in the equation are pulled [35,37]. One of the simple
results is that when f(u)=u 6 f (0) the fronts are pulled, which con3rms our intuitive understanding
that for a front to become pushed one needs the nonlinearities to enhance the growth. For (sets of)
equations that admit uniformly translating fronts one typically 3nds the same trend, namely that the
enhancement of the growth by the nonlinear terms in the dynamical equations tends to favor the
occurrence of pushed fronts.
It is important to realize, however, that for coherent and incoherent pattern forming fronts,
these intuitive ideas do not necessarily apply. Several of the examples to be discussed in
Section 2.11 will illustrate this: in the quintic CGL equation the cubic term enhances the growth,
but there are large regions of parameter space where nonlinear dispersion eDects completely suppress the occurrence of pushed fronts. Likewise, fronts in the Kuramoto–Sivashinsky equation of
Section 2.11.4 are pulled, but if one adds a linear term to the equation, a pulled to pushed transition is found. However, in dynamical pattern forming equations which derive from a Lyapunov
functional and for which one has an reasonable understanding of whether the nonlinearities enhance
or suppress the growth about the unstable state, one’s intuition of what to expect is usually correct. Our discussion of front dynamics in the Swift–Hohenberg and Cahn–Hilliard equation will
illustrate this.
As the example discussed at the end of Section 2.7.1 illustrated, for the second order partial F-KPP
equation, the pushed front solutions can in a number of cases be explicitly constructed analytically.
The reason is the following. While for an arbitrary v the large-- asymptotics (58) is characterized by
two diDerent exponentials, the pushed front solutions (60) of the F-KPP equation are characterized
by one root. It is therefore possible (but not necessary) that the pushed solutions obey a 3rst order
equation. The “reduction of order method” brieKy reviewed in Section 2.7.1 is based on substituting
a 3rst order Ansatz into the second order ordinary diDerential equation for the uniformly translating
solutions. The most remarkable success of this method is that the pushed solutions of the quintic
CGL equation have been found this way—see Section 2.11.6.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.7.6. Precise determination of localized initial conditions which give rise to pulled and pushed
fronts, and leading edge dominated dynamics for non-localized initial conditions
So far, we have focused on the front dynamics emerging from initial conditions which are “steep”,
in the sense that they fall oD faster than exp[ − ∗ x]. This is because only for such initial conditions
does the front selection problem have a sharp and unique answer—whether fronts are pushed or
pulled is an inherent property of the dynamical equations. Indeed, the values of the corresponding
velocities v∗ and v† are determined completely by the equation itself.
What happens if the initial conditions are not steep, i.e., fall of slower than exp[ − ∗ x]? Given
our assumption of locality of our dynamical equations (no inKuence oD from points arbitrarily far
away) the answer lies hidden again in our analysis of the linear dynamics illustrated in Fig. 5. If
initial conditions fall oD with steepness ¡ ∗ , the spatially slowly decaying leading edge on the
right, which moves with velocity venv (), expands in time. In other words, in the frame moving
along with this leading edge, the crossover region to the slower part of the pro3le recedes to the left
and larger and larger parts of the pro3le move with asymptotic velocity venv . Whatever the nonlinear
dynamics is, it must be compatible with this dynamical constraint in the leading edge of the pro3le.
Indeed, such initial conditions which are not steep necessarily lead to a front moving with speed
venv () with ¡ ∗ , unless a pushed front solution exists whose speed v† is larger than venv . For,
if such a pushed front solution with v† ¿ venv () exists, this solution will invade the leading edge
according the mechanism sketched in Fig. 8(b). Thus we conclude:
For an equation whose fronts are pulled, all initial conditions with steepness ¡ ∗ lead to
fronts moving with speed venv () ¿ v∗ . In other words, in the pulled regime we can identify
“localized” initial conditions which lead to pulled fronts with initial conditions which are steep,
i.e., which fall o> faster than exp[ − ∗ x].
For an equation whose fronts are pushed, only initial conditions with steepness ¡ 1 lead to
fronts moving with speed venv () ¿ v† , where 1 is the smallest root for which venv (1 ) = v† .
In other words, localized initial conditions which lead to pushed fronts are initial conditions
which fall o> faster than exp[ − 1 x], where 1 is determined implicitly by the pushed front
solution through the requirement venv (1 ) = v† .
Intuitively the mechanism through which non-localized initial conditions lead to fronts that move
faster than the naturally selected pulled or pushed speed, is very much like “pulling along” the
nonlinear front. However, in order to distinguish them from the pulled front solutions which naturally
emerge from all localized initial conditions, we refer to this type of dynamics more generally as
“leading edge dominated” dynamics [144].
2.7.7. Complications when there is more than one linear spreading point
In our discussion of the pushed fronts, we have so far assumed that the linear dispersion is such
that the venv () versus diagram is of the type sketched in Fig. 3, i.e., that there is only one branch
which in the limit ↓ 0 corresponds to a positive growth rate !i and hence to a divergent venv ().
This is the normal situation for problems where there is essentially one branch of linear modes which
is unstable. We now brieKy discuss the subtleties associated with having more than one unstable
branch of the dispersion relation.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
71
(b)
Fig. 11. Illustration of possible venv () versus diagrams in the case in which there is more than one unstable branch of
modes. The relevant linear spreading velocity in each case is indicated with a 3lled circle. As explained in the text, the
asymptotic steepness of a pushed front has to be associated with one of the two dashed branches.
Two examples of possible venv () diagrams for the case of two unstable branches are sketched
qualitatively in Fig. 11. As we discussed before, according to the fully linear spreading analysis,
the relevant linear spreading velocity v∗ is the largest spreading velocity of the two branches. The
crucial point to understand for the proper extension of the concept of a pushed front in such cases is
that the linear dynamics associated with the two individual branches is completely independent—we
can repeat the earlier discussion of how pushed fronts can emerge for each branch individually,
and this leads to the conclusion that a pushed front is again a front whose asymptotic exponential
∗
decay is faster than e− - . However, in addition the exponential steepness † of the pushed front
should be larger than the value 1∗ or 2∗ associated with the minimum of the branch to which it
belongs [86]. 35 In other words, the leading steepness of a pushed front should correspond to 36 one
of the two dashed branches in the 3gure. In the case of Fig. 11(a), a uniformly translating pushed
front solution is a solutions whose asymptotic decay is as e−2 - , not as e−1 - . Likewise, for the case
sketched in Fig. 11(b) a transition from the pulled front with velocity v∗ to a pushed one, whose
asymptotic behavior is as e−2 - , could occur.
Like before, the possibility of such situations to occur is intimately related with counting arguments for the multiplicity of front solutions. E.g., for uniformly translating fronts with a dispersion
relation consistent with one of the cases sketched in Fig. 11, one in general expects there to be a
two-parameter family of front solutions. For the case of Fig. 11(a) pushed front solutions with the
required asymptotic behavior are then expected to come as isolated front solutions, just as before.
35
Another way to arrive at the same conclusion, stressed in particular by Chomaz and Couairon [86], is to note that
asymptotic behavior with steepness given by one of the two left branches 1 or 2 can only emerge dynamically from
non-localized initial conditions falling oD with this steepness: Chomaz and Couairon [86] call these branches non-causal.
In the language of [421] one can understand this from the fact that on these branches the group velocity is less than the
envelope velocity—perturbations cannot work their way up towards the far tip.
36
In the case of coherent pattern forming front solutions, “correspond to” should be read as “analytically connected to”,
since as we explained in Section 2.7.2 the asymptotic decay of a pushed front does not correspond to a value on the
dashed branch of venv .
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.8. Relation with existence and stability of front stability and relation with
previously proposed selection mechanisms
In this section, we brieKy discuss the relation between front selection and stability, and the relation
with some of the older proposed selection mechanisms.
2.8.1. Stability versus selection
The main diDerence in perspective with the “marginal stability” selection mechanism of the 1980s
[38,111,420,421] is that we here emphasize that by starting from the classical linear spreading
analysis [49,62,264] the concept of a linear spreading velocity v∗ naturally arises. This concept
holds irrespective of whether or not there are well-de3ned front solutions, and applies equally well
to partial diDerential equations, diDerence equations and integro-diDerential equations. From this
perspective, the possibility of having pulled fronts arises most naturally and independently of whether
they are coherent or incoherent. In addition, the saddle point integration immediately shows that the
intermediate asymptotic dynamics giving the approach to the asymptotic spreading speed is that of
a diDusion equation, even if the equation itself is not at all a simple diDusion equation. This latter
fact lies at the basis of the universal relaxation behavior discussed in the next section. Moreover, by
exploring the constraints imposed by the linear dynamics, the essential properties of pulled fronts
follow.
The drawback of the “marginal stability” formulation of [420,421] is that when one wants to relate
selection with the stability properties of fronts, one can at best only understand the selection once one
is already close to an asymptotic front solution. 37 Why the intermediate asymptotics bring one there
is less clear—even though the two are quite consistent the attempt in the marginal stability approach
to treat selection and linear spreading in one fell swoop makes the problem unnecessarily cumbersome
and masks the generality of the linear spreading concept. Likewise, the universal relaxation behavior
of pulled fronts that we will discuss in the next section is virtually impossible to get from a stability
analysis (for uniformly translating or coherent pulled fronts, the spectrum of the stability operator is
continuous with arbitrarily small eigenvalues), whereas it naturally emerges from the linear spreading
concept.
Our simple observation that whenever a solution (60) or (76) with v† ¿ v∗ exists whose asymptotic steepness is larger than ∗ , this solution can and will invade the leading edge and lead
to pushed front propagation, is actually a general formulation which encompasses the “nonlinear
marginal stability” scenario that when one considers fronts as a function of there velocity, front
solutions with speed v ¡ v† are unstable [38,421]. One can see this as follows [144, Appendix I].
For dynamical equations which are translation invariant, the translation mode is always a mode with
zero eigenvalue of the stability operator. At v = v† the front solution is a strongly heteroclinic orbit,
and hence the zero eigenmode of the stability operator corresponds to a strongly heteroclinic orbit.
Continuity and counting arguments then imply that for v close to but diDerent from v† , there is a
strongly heteroclinic solution of the stability operator whose eigenvalue crosses zero at v = v† . In
other words, when the stability of the front solutions is studied for varying velocity v, the stability
37
Booty et al. [53] have also analyzed the dynamics in the leading edge of the F-KPP equation. Conceptually, their
approach has much in common with that of [420,421].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
73
changes at v† —front solutions with velocity v ¡ v† are unstable due to the invasion of the pushed
front solution into the leading edge.
Another reason for separating front stability from front selection is that the simulations discussed
in Section 2.11 will show that the state generated by the front becomes absolutely unstable in the
frame moving with the front, this entails a transition from a coherent pattern forming front to an
incoherent pattern forming front, not necessarily a change from pulled to pushed dynamics.
2.8.2. Relation between the multiplicity of front solutions and their stability spectrum
The above discussion of the implication of the existence of a pushed front solution for the stability
of fronts also illustrates that there generically is an intimate connection between the multiplicity of
uniformly translating and coherent front solutions and the stability properties. We can illustrate this
for the other stability modes as follows. As we discussed in the previous section, a uniformly
translating or coherent front solution corresponds to an orbit in the phase space of the ordinary
diDerential equations that govern these solutions, and the multiplicity of these solutions is determined
by “counting arguments” for the dimension of the manifolds that Kow into and out of the 3xed points
that correspond to the asymptotic states. Consider now the case in which the dynamical equations
admit a one-parameter family of front solutions, parametrized by their speed v—we argued that
this is the usual case for front propagation into unstable states. If we now pick an arbitrary front
solution from this family and write down the stability operator for perturbations about the front
solution, then the counting arguments for the stability modes is essentially unchanged because the
linearization about the asymptotic states before and after the front is unchanged. Hence the existence
of a one-parameter family of fronts, parametrized by v, generically implies that if we 3x v and
consider the stability modes, then we expect there to be in general a one-parameter family of stability
modes, parametrized by their growth/decay rate. Together with the fact that translation invariance of
the dynamical equation implies that there is a zero mode of the stability operator, this also implies
that there generically is a continuous spectrum of the stability operator near zero. A general analysis
of the asymptotic behavior of these modes about the unstable state shows that the modes from this
spectrum which decay faster than the front solution are stable, and that those which decay less fast
than the front solution are unstable [421]. This line of analysis in combination with the one above
for the possible presence of a localized stability mode if a pushed front solution exists, gives a
quite complete generic picture of the stability of the front solutions in the generic case. In short:
if no pushed front solution exists, so that the selected fronts are pulled, then the front solutions
with v ¿ v∗ are stable to perturbations which decay faster than the front itself, and unstable to those
which spatially decay less fast than the front itself. If a pushed front solution exists, then the generic
picture is that front solutions with v ¡ v† are unstable to the localized mode, and those with speed
v ¿ v† are stable to this mode and to perturbations whose spatial decay to the right is faster than
that of the front solution itself.
For coherent pattern forming fronts, similar arguments apply. Let us again focus on the generic
case (see Section 2.7.2) that these front solutions come into a two parameter family. If we consider
a particular front solution at a 3xed velocity v, then the generic scenario that results from similar
continuity arguments is that any arbitrary front solution which in the tail does not match up with the
maximal growth rate !i in the leading edge, or, in other words, whose asymptotic behavior does not
correspond with the left branch drawn with a full line in Fig. 3(a), is unstable. This is the reason we
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
focused on the analysis of venv de3ned in Eq. (21) in Section 2.2: only that branch matches up with
asymptotic coherent front solutions which are stable to perturbation that decay spatially faster to the
right than the front solutions themselves. The discussion of stability of a possible localized stability
mode, associated with the existence of a pushed front solution, is analogous to the one given above.
The explicit stability calculation for the uniformly translating fronts in the F-KPP equation or
extensions of it can be found in a number of papers [60,72,144,149,216,230,252,317,371] and will
not be repeated here. The results are completely in accord with the above general discussion.
2.8.3. Structural stability
We 3nally comment brieKy on the proposal to connect propagating front selection with “structural
stability” ideas. According to this conjecture [83,334], the pulled front is the natural front speed as it
is the only front solution which is “structurally stable” to small changes in the dynamical equations
(like those which would suppress the instability or make the state = 0 even linearly stable). It
is easy to convince oneself that uniformly translating pulled solutions do have this property, as the
dynamically relevant front solutions are characterized by a real spatial decay rate only, but that as we
discussed at the end of Section 2.7.2 coherent pattern forming front solutions which are characterized
by a decay rate and wavenumber in the leading edge, generically do not have this property. Indeed,
the quintic CGL equation provides an explicit counterexample to the “structural stability” postulate:
As we shall see under (iii) in Section 2.11.6, for the quintic CGL equation the selected wavenumber
can jump at the pulled to pushed transition.
2.8.4. Other observations and conjectures
The issue of front selection issue has intrigued many authors, so various other observations and
conjectures have been made. In Appendix B we brieKy discuss some of these: a (wrong) conjecture
concerning the analytic structure of pushed front solutions, an observation about obtaining the selected
uniformly translating front by studying the front solutions on a 3nite interval, and the connection
with Hamilton-Jacobi theory and renormalization group ideas.
2.9. Universal power law relaxation of pulled fronts
Up to now, we have focused on the asymptotic front velocity. Let us now assume we study an
equation whose fronts are pulled, and ask how the asymptotic front velocity is approached. To do so,
we 3rst have to state how we de3ne a time-dependent velocity of a front during the transient regime
when it approaches its asymptotic value. We will be quite pragmatic in our discussion: the asymptotic
convergence to v∗ will turn out to be very slow, so slow that the diDerences between various
conventions do not really matter. We will therefore focus simply on the most natural de3nition.
In this section, we will simply state the results for the convergence to v∗ ; the essential ingredients
of the derivation are reviewed in Section 4.
It is good to stress that the universal relaxation only holds for pulled fronts. The discussion of
Section 2.8 implies that the stability spectrum of a pushed front solution is gapped, and hence that
a pushed front relaxes exponentially fast to its asymptotic velocity and shape [144,227,421]. This is
in line with the intuitive idea illustrated in Fig. 8 that a pushed front invades the region ahead of it
with a 3nite speed.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
75
2.9.1. Universal relaxation towards a uniformly translating pulled front
If a pulled front is asymptotically uniformly translating, we can simply follow the same idea as
in our discussion of the linear spreading problem in Section 2.1: In essence our convention will
always be to determine a front speed vC (t) by tracking the position xC (t) of the level line where
the dynamical 3eld reaches a level C; the velocity is then the speed of this point
d xC
:
(77)
(xC ; t) = C ⇔ vC (t) =
dt
Of course, for dynamical 3elds with more than one component, the transient velocity could in principle depend on which component we track, but we will not distinguish this possibility notationally,
as we will focus on those aspects which are independent of those details.
The analysis [144] of the convergence of the nonlinear pulled front speed to v∗ is based on
matching the behavior in the leading edge, where corrections to the asymptotic exponential behavior
∗
e− are governed by an equation which in dominant order is a diDusion-type equation, to the
behavior in the fully nonlinear region. In this step, the fact that the asymptotic front solution ,v∗ ()
∗
has according to (59) a e− behavior plays a crucial role—see Section 4. The 3nal result of the
matching analysis [144] is the following exact expression for the velocity v(t) of an asymptotically
uniformly translating front,
√
3
1
3 v(t) = v∗ − ∗ + √
(78)
+O 2 ;
2 t 2 D(∗ )2 t 3=2
t
which holds provided one starts from steep initial conditions which fall oD faster than e− x . 38 This
slow power-law like relaxation of the velocity to its asymptotic value v∗ entails a slow relaxation of
the front pro3le to its asymptotic shape. Indeed, if we de3ne X (t) as the shift of the front position
in the frame moving with speed v∗ ,
t
3
dt (v(t ) − v∗ ) ⇔ X (t) − ∗ ln t + O(t −1=2 ) ;
(79)
X (t) =
2
∗
then one 3nds for the relaxation of the front pro3le to its asymptotic shape
√
1
(x; t) ,v(t) (X ) + O 2 ; X Dt ;
t
(80)
where X is the frame
X ≡ − X (t) = x − v∗ t − X (t)
(81)
which includes the logarithmically increasing shift X (t).
There are many points to note about these universal relaxation results for pulled fronts and the
physical picture underlying their derivation:
(i) While the above results are exact for all pulled fronts which asymptotically are uniformly
translating, for the special case of the F-KPP equation, the 1=t term was known for over
38
To be more precise, the result holds provided the initial conditions fall oD exponentially faster than exp[ − ∗ x], i.e.,
∗
∗
provided there is some ¿ 0 such that (x; t = 0)e x ¡ e−x as x → ∞. The special case in which (x; t = 0) ∼ x4 e− x
is discussed in [54,144,255,256,314].
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
1.0
t=0
t=5
t=15
t=10
u 0.5
u
0.0
165
u
170
∗
Φ (ξ)
175
u
180
∗
Φ (ξ)
185
u
190
∗
Φ (ξ)
195
200
x
Fig. 12. Illustration of the logarithmic shift of the transient pro3le relative to the asymptotic pro3le ,v∗ moving with
constant speed v∗ , for the case of the F-KPP equation with f(u) = u − u3 . The solid line shows the actual shape and
position of the front emerging from an initial condition at time t = 0 at three successive times. The dotted line shows
the uniformly translating solution ,v∗ (labeled ,∗ in the 3gure) at three successive times; at t = 5 this solution was
placed such that it intersects the actual solution at u = 0:5. The thick solid line indicates that the transient solution and
the asymptotic solution separate more and more as time goes on, even though the shape of the transient front is always
close to the asymptotic one. From [144].
20 years [138,139,291] and was derived rigorously in 1983 by Bramson [54]. Since then it has
been re-derived by various methods (including a matching analysis of [64] similar to ours),
but to my knowledge even for the F-KPP equation the universal 1=t 3=2 term was not known
before the matching analysis [144] described in more detail in Section 4.
(ii) The dominant logarithmic shift in X (t) which is driven by the diDusive dynamics in the
leading edge implies that when viewed in the frame moving with the asymptotic velocity v∗ ,
the front moves back while it is relaxing. This is illustrated in Fig. 12 for a simulation of
the F-KPP equation starting from a localized initial condition. The plot shows the actual front
pro3le at times t = 5, 10 and 15 (full line), and compares these with the uniformly translating
asymptotic pro3le ,v∗ (dotted line) which has been placed so that it intersects the transient
pro3le at =1=2 at time t =5. The increase of the length of the thick line with time illustrates
that the actual transient pro3le and the asymptotic pro3le ,v∗ separate more and more in time.
(iii) The results which are illustrated in Fig. 12 show quite clearly that any method based on
linearizing about the asymptotic front solution ,v∗ will not work: the diDerences between the
actual front solution and this asymptotic one grow arbitrarily large! This is why the insight
to use the logarithmically shifted frame X is crucial for any theoretical analysis.
(iv) Nevertheless, the 3gure does con3rm that already quite soon the transient front shape is close
to the asymptotic shape ,v∗ . This fact is expressed in precise mathematical terms by (80):
to order t −2 the front shape is given by the expression for the uniformly translating front
solution ,v , provided we use the instantaneous value v(t) given by (78) for the velocity, and
put the front at the appropriately shifted position X . In other words, to O(t −2 ) the transient
front shape follows the uniformly translating solutions adiabatically. This fact was 3rst noted
empirically in simulations by Powell et al. [354].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
77
(v) The above results are valid for any pulled front which asymptotically is uniformly translating:
it is independent on the precise nonlinearities and applies to all pulled fronts in equations
for which the linear spreading speed v∗ can be determined according to our discussion of
Section 2.4. The fact that these results apply equally well to diDerence equations like the 3nite
diDerence version (45) of the F-KPP equation or the diDerence-diDerential equation discussed
in Section 3.23, is that the saddle point expression for v∗ ensures that the corrections to the
∗
asymptotic exponential e− x behavior are governed by a diDusion-type equation. In other
words, as these corrections become arbitrarily smooth for long times, the lattice eDects only
give higher order corrections [144,146]. See Section 4 for further details.
(vi) Since the above results are independent of the precise form of the nonlinearities in the dynamical equation as long as the fronts are pulled, one may wonder where the nonlinearities
are hidden. Comparison with the result for the convergence to v∗ according to the fully linear dynamics discussed in Section 4.1 shows that the prefactor 3/2 of the 1=t term and the
subdominant t −3=2 term both reKect the nonlinear behavior. 39
(vii) In line with the earlier conclusion of Section 2.3 that the limits t → ∞, 3xed and → ∞, t
3xed do not commute, we stress
√ that the expression (80) is the correct asymptotic expression
for t → ∞, X 3xed. At X Dt, there is a crossover to a diDerent asymptotics that governs
the large X limit [144].
(viii) Note that the above result for the universal velocity relaxation holds independent of the initial
conditions and of the level line which is used to track the position of the front, provided one
is not at the pulled to pushed transition. 40 In fact, the correction to the velocity relaxation
associated with the shape relaxation is according to (80) of order [,v (-)=v] dv(t)=dt ∼ t −2 .
(ix) The terms displayed in (78) for v(t) and in (80) for the relaxation of the front pro3le are
also the only universal terms. This can easily be seen as follows. Suppose we compare the
velocity formula for two cases, one starting from some steep initial condition at time t = 0
and the other one by viewing the dynamical state at time t = Wt as the initial condition. For
large times the 1=t terms for the two cases diDer of order Wt=t 2 . Thus the term of order 1=t 2
depends on the initial conditions, i.e., is non-universal.
(x) Formula (78) shows that the asymptotic velocity is always approached from below. This
explains why in many 3nite time simulations of pulled fronts the published velocity data
are slightly below v∗ (we will encounter several examples of this later). This fact together
with Eq. (80) for the shape relaxation of the front imply that for the transient dynamics the
front solutions with speed v ¡ v∗ are important. This is the reason for our cautionary note in
Section 2.2 about using the phrase “the minimum velocity”.
(xi) Extensive numerical investigations and illustrations of the universal relaxation behavior of
uniformly pulled fronts can be found in [144,146]. An example of such tests will be discussed
in Section 4.2.
39
As we will show in Section 4.1, Eq. (203), according to the fully linear dynamics a level line asymptotically recedes
as −(ln t)=2∗ in the frame moving with v∗ , rather than as the −3(ln t)=2∗ term of the nonlinear front pro3le. Hence if
we would also draw in Fig. 12 the pro3le as it evolves according to the linearized equation, it would for large times and
not too large u lie in between the asymptotic front pro3le ,v∗ and the actual transient pro3le.
40
At the pulled to pushed transition the prefactors of the 1=t and 1=t 3=2 term are diDerent—see Appendix G of [144].
78
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.9.2. Universal relaxation towards a coherent pattern forming pulled front
How to de3ne the instantaneous front velocity for a coherent pattern forming fronts is subtle issue.
If one traces the position of the foremost point where a dynamical 3eld which develops oscillations
reaches a given value, this position will make 3nite jumps when a new oscillatory part in the
leading edge grows large enough that it reaches the predetermined level C. Since the time between
successive jumps will be 3nite, averaging over some 3nite time then already gives a crude idea of
the slow long-time convergence. A better way is to determine numerically an envelope of the front
pro3le from traces of e.g. the maxima of the oscillations during one period, and to then determine
the velocity from the positions of a level line of this empirical envelope of the front [388,383]. The
advantage of this method is that it also works well when one wants to trace level lines in the range
where the nonlinearities in the dynamical equation are clearly important. If, on the other hand, one
decides to track the front velocity in the leading edge only, then a good method is to 3t a decaying
oscillatory exponential to the front pro3le, and to determine the position of a given level from that
3t. We will not dwell on the advantages and disadvantages of the these methods further, as they are
of little relevance when one wants to probe the universal slow long-time relaxation.
Under the same assumptions as before for the uniformly translating fronts (see footnotes 38 and
40), the front velocity of a coherent pulled front relaxes to its asymptotic value v∗ as [147,383,388]
√
1
3
1
3 √
v(t) ≡ v∗ + Ẋ (t) = v∗ − ∗ +
;
(82)
+
O
Re
2
2 t 2(∗ )2 t 3=2
t
D
which reduces to the result (78) for uniformly translating fronts when Im D = 0. The relaxation of
the front pro3le to its asymptotic behavior (56) is characterized by the relaxation of the velocity
and a global phase 5(t): it is found that in this case one can write for the long-time asymptotics
1
−in[/t+5(t)] n
(x; t) ≈
(83)
e
,v(t) (X ) + O 2 ;
t
n=0;±1;:::
where 5(t) is given by [147,383,388]
√
1
1
3 ∗
5̇(t) = −kr Ẋ (t) − ∗ 3=2 Im √ + O 2 :
2 t
t
D
(84)
These equations simply express that to O(t −2 ) also the pattern forming fronts follow adiabatically a
family of coherent front solutions.
Essentially all the remarks made about the slow relaxation behavior of uniformly translating fronts
apply equally well to the relaxation behavior of the pattern forming fronts—the only additional
feature is the analogous slow convergence of the frequency 5̇(t): it implies a slow convergence of
the wavelength of the pattern generated by the front. To our knowledge, this behavior has been
veri3ed both qualitatively as well as quantitatively only for the Swift–Hohenberg equation—see
Section 2.11.2 and Fig. 19 below. It does not appear to have been studied experimentally in a
systematic way, although the slow velocity relaxation of pattern forming fronts does appear to play
a role in some pattern forming experiments—see Section 3.1.
2.9.3. Universal relaxation towards an incoherent pattern forming pulled front
Quite remarkably, though we cannot make a prediction for the front shape relaxation for an
incoherent pulled front, we will argue in Section 4 that the same velocity relaxation formula (82)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
79
applies to an incoherent pattern forming pulled front. The reason for this is that even when a pulled
front is nonlinearly incoherent, the behavior in the leading edge is still very smooth and coherent, as
it is governed by the same linear dynamical equation for the expansion about the linear spreading
point. This latter observation also implies that the velocity of an incoherent front is in practice most
easily measured in the leading edge. Even the phase correction 5(t) is well-de3ned there and follows
(84)—the results for the quintic CGL equation shown in Fig. 19 provide evidence in support of
this claim.
2.10. Nonlinear generalization of convective and absolute instability on the basis of the results
for front propagation
In Section 2.5 we discussed the diDerence between convective and absolute instabilities in the case
that the evolution of a perturbation around the unstable state evolves according to the linearized
equations. As Fig. 7 illustrated, if we focus on the right Kank of the local perturbation, the linear
instability is absolute if v∗ ¿ 0 and convective if v∗ ¡ 0—in the latter case the perturbation “retreats”
so that it eventually dies out at any 3xed position. If perturbations have enough time to grow that
the nonlinearities become important and that a nonlinear front develops, it is straightforward to
extend the distinction between the two regimes to the nonlinear case [85,99,412]. Clearly, in the
pulled regime the criterion is the same since the nonlinearities do not aDect the asymptotic front
speed, so the instability is nonlinearly convectively unstable for v∗ ¡ 0 and nonlinearly absolutely
unstable for v∗ ¿ 0. When the fronts are pushed in the dynamical equation under consideration,
the criterion becomes simply that the instability is nonlinearly convectively unstable if v† ¡ 0 and
nonlinearly absolutely unstable when v† ¿ 0. That’s all there is to it. A recent nice experiment on the
Kelvin–Helmholtz instability in which the transition from a nonlinearly convective to a nonlinearly
absolute instability was observed, can be found in [186].
2.11. Uniformly translating fronts and coherent and incoherent pattern forming fronts in
fourth order equations and CGL amplitude equations
In this section we illustrate many of the basic issues of front propagation into unstable states
by reviewing the diversity of such fronts in Complex Ginzburg Landau (CGL) equations and in
a number of well known fourth order partial diDerential equations which have been introduced in
the literature as model problems for a variety of physical phenomena. 41 In fact, we encounter all
three diDerent modes of front propagation dynamics both for pushed and for pulled fronts in these
equations, so taken together the examples discussed below give a good idea of the richness of front
propagation into unstable state as well as of the power of the concepts of a pulled and pushed fronts.
We will limit our discussion completely to the dynamics of fronts which propagate into an unstable
state, so it is important to keep in mind that such behavior constitutes just a small fraction of the wide
41
We focus in the section on rather well known model equations. From a historical perspective, it is interesting to
note that pulled propagating fronts were also found back in 1983 in a fourth order caricature model aimed at describing
sidebranching in dendrites [248]. The dispersion relation for the sidebranch instability in this model is of the same form
as the general form we consider here, Eq. (86) below.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
range of dynamical behavior that is found in these equations. The books by Greenside and Cross
[189] and by Nishiura [320] provide nice complementary introductions to these model equations.
While the nonlinear front dynamics will be found to be diDerent for all of the fourth order equations
we will discuss, the linear dynamics obtained by linearizing the equations about the unstable state
will always be a special case of
9t = a − b92x − 94x :
(85)
The negative prefactor of the fourth order derivative term is required to ensure stable behavior
of the equations at the short length scales. Indeed, the dispersion relation corresponding to this
equation is
! = i(a + bk 2 − k 4 ) ;
(86)
so for any value of a and b the short wavelength modes at large k are always damped (!i ¡ 0).
The requirement that the state about which we linearize is unstable for a range of modes implies
that the coeRcients should obey a + b2 =4 ¿ 0. It is simply a matter of some straightforward algebra
to work out the linear spreading√point equations (12) and (13) explicitly. We 3nd that there are two
regimes: for a ¿ 0 and b ¡ − 12 a we get
2
∗
b2 − 12a)(−b − b2 − 12a)1=2 ;
v4th
ord (a; b) = √ (−2b +
3 6
1
∗
√
4th
(−b
−
(a;
b)
=
b2 − 12a)1=2 ;
ord
6
kr;∗4th ord (a; b) = 0 ;
D4th ord (a; b) = b2 − 12a ;
(87)
√
√
while for a ¿ 0 and b ¿ − 12 a as well as for a ¡ 0 but b ¿ 2 −a
2
∗
7b2 + 24a)(−b + 7b2 + 24a)1=2 ;
v4th
ord (a; b) = √ (2b +
3 6
1
∗
√
4th
(−b
+
(a;
b)
=
7b2 + 24a)1=2 ;
ord
2 6
1
kr;∗4th ord (a; b) = ± √ (3b + 7b2 + 24a) ;
2 2
D4th ord (a; b) = 2 7b2 + 24a ;
(88)
√
As we shall see below, in the 3rst regime a ¿ 0, b ¡ − 12 a, fronts are uniformly translating,
while in the other regime they are pattern forming.
2.11.1. The extended Fisher–Kolmogorov equation
While the relevant asymptotic front solutions of the F-KPP equation are uniformly translating
and monotonically decaying in space to the right, if one includes a fourth order derivative term
in the equation, the front dynamics becomes much more rich. If we consider the case in which
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
EFK equation
81
EFK equation
uniformly translating pulled front at γ = 0.06
pattern forming pulled front at γ = 0.25
20
20
15
15
10
10
5
5
0
0
0
40
80
120
(a)
160
0
40
80
120
160
(b)
Fig. 13. Space–time plot of simulations results for pulled fronts in the EFK equation (89). The lines denote the front
position at successive time intervals of 1.5, and are shifted upward relative to each-other for clarity. The initial condition
is a Gaussian of amplitude 0.1 centered at the origin and the total simulation time is 75. (a) Illustration of the front
dynamics in the regime 2 ¡ 1=12. The plot, made for 2 = 0:06 illustrates that the fronts converge to uniformly translating
∗
solutions propagating with velocity vEFK
. (b) The simulation for 2 = 0:25 illustrates that in the regime 2 ¿ 1=12 the pulled
fronts generate patterns: they leave behind an array of kinks. As can be seen from the graph, nodes are conserved under
the dynamics, and as a result the wavelength of the pattern is given by the node conservation formula (91). Note also the
smooth uniform propagation of the leading edge, in spite of the complicated nonlinear dynamics in the front region itself.
the nonlinear term is a simple cubic nonlinearity so as to study pulled fronts, we arrive at what is
usually referred to as the Extended Fisher–Kolmogorov or EFK equation [113,421]
9t u = 92x u − 294x u + u − u3
(2 ¿ 0) :
(89)
This form of the equation is the most suitable for studying the behavior in the F-KPP limit 2 → 0.
For any nonzero 2, however, upon transforming to the scaled coordinate x = x=21=4 the linear terms
√
in this equation are precisely of form (85) with a = 1 and b = −1= 2. Thus we can use the above
expressions provided we rescale all lengths by a factor 21=4 ; e.g., the linear spreading speed of the
EFK equation is simply
√
∗
∗
vEFK
= 21=4 v4th
(90)
ord (1; −1= 2) :
The most interesting aspect of the EFK equation as far as front propagation is concerned is the
bifurcation that occurs in the pulled front behavior at 2 = 1=12: For
√ 2 ¡ 1=12, we are in the regime
where the pulled front parameters are given by (87) as b ¡ − 12, and hence where we have
kr∗ = 0. This means that the asymptotic front pro3le falls oD monotonically, just like in the F-KPP
equation. Fig. 13(a) shows an example of the front dynamics in this regime. For 2 ¿ 1=12, however,
kr∗ = 0, and the leading edge of the pulled front falls oD in an oscillatory manner. The dynamics
that results from this oscillatory behavior is illustrated in Fig. 13(b): the oscillations in the leading
edge periodically grow in size and “peel oD” from the leading edge to generate an array of kinks
between the two stable states u = ±1 [113].
In the numerical simulation one empirically observes that, in this process, nodes in the pro3le
where u goes through zero, never disappear nor are generated spontaneously: They are formed in the
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
far leading edge, and then just gradually drift through the leading edge until they detach and come
to rest as a kink. To our knowledge, there is no real theory which shows under what conditions
or for what type of equations nodes are “conserved” under the dynamics. However, given that we
empirically observe it, we can calculate the wavelength of the stationary nonlinear kink pattern
behind the front [111,113]! In the frame moving with the asymptotic speed v∗ , the oscillatory
part of the pro3le is for long times given by exp[ − i(!r∗ − kr∗ v∗ )t], so the angular frequency with
which nodes pass any 3xed position in the leading edge is |!r∗ − kr∗ v∗ |. Far behind the front, the
kinks come to rest in the x frame. If the wavenumber of the kink pro3le is qstat , then the angular
frequency with which nodes pass a 3xed position far behind the front in the moving frame is
simply v∗ qstat . Node conservation together with expressions (88) therefore implies that
√
3 (3b + 7b2 + 24a)3=2
∗ ∗
∗
√
qstat = |!r =v − kr | = √
:
(91)
8 2 2b + 7b2 + 24a
The expression for the front speed in both regimes, as well as this one for the pattern wavenumber in
the regime 2 ¿ 1=12, was found to be fully consistent with the results of numerical simulations. 42
In fact, an ideal periodic kink pattern of the type generated behind the front is not necessarily
stable, so the emerging near-periodic state is usually only an intermediate asymptotic state: For
1=12 ¡ 2 ¡ 1=8, successive kinks and antikinks attract each other. This attraction is very weak for
widely spaced kinks, and so the bunching instability that it gives rise to is virtually unnoticeable in
the simulations. For 2 ¿ 1=8, the spatially oscillatory tail of a single kink gives rise to an oscillatory
interaction between kinks [113,340]. This is just one example of the rich type of behavior that the
EFK displays—see [340,410] for further details or [370,371] for work on the existence and stability
of the front solutions.
Just like the F-KPP equation exhibits a transition to pushed fronts if the cubic nonlinearity u3 is
changed into a nonlinear function f(u) whose growth is enhanced over the linear term at intermediate
values (e.g. by a quadratic nonlinearity u2 ), so does the EFK equation [421]—we already illustrated
this in Fig. 10. Even for 2 ¡ 1=12, when pulled fronts in the EFK equation are spatially monotonic,
pushed fronts in this regime can be non-monotonic even though they remain uniformly translating,
see Fig. 10 and [421]. So, as we already stressed before, unlike fronts in the F-KPP equation, both
pulled and pushed fronts in the EFK equation can be non-monotonic—monotonicity has nothing to
do with front selection.
What else does the EFK equation teach us? Well, this equation with its bifurcation from uniformly
translating fronts for 2 ¡ 1=12 to pattern forming fronts for 2 ¿ 12 is probably the simplest equation
that illustrates the essence of pulled dynamics: the dynamics in the leading edge just does what the
linear spreading point conditions impose, while the rest of the front where the nonlinear saturation
term becomes important just cannot do anything but adjust to the dynamics enforced by the leading
edge dynamics. In the present case, where essentially the only basic coherent structures admitted by
the equation are kinks, the state behind the front is in 3rst approximation an array of such kinks or
domain walls. This state itself may be unstable or have other nontrivial dynamics, but this does not
really aDect the propagation of the pulled front.
42
∗
Inspection of the data points for the velocity in [113] shows that they fall slightly below the predictions for vEFK
. This
is due to the fact that in a 3nite time simulation one always observes a velocity slightly less than v∗ due to the power
law convergence to v∗ from below discussed in Section 2.9. Similar eDects can be observed in virtually all published data
of studies of pulled fronts, some of which are reviewed in Section 3.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
83
As we discussed in Section 2.7, the coherent pattern forming front solutions (56) are coherent
in that they are temporally periodic in the frame moving with velocity v∗ . The node-conservation
argument is one immediate implication of this. Hence the empirical observation of node-conservation
in the EFK equation for 2 ¿ 0 strongly suggests that the pattern forming solutions are coherent, i.e.,
of the form (56). Indeed one can convince oneself [429] term by term in the expansion that the
EFK equation for 2 ¿ 1=12 should admit a two-parameter family of front solutions, parametrized by
their velocity v and the wavelength of the kink pattern behind the front.
We 3nally note that while the EFK equation has been introduced mainly as a model equation to
illustrate the transition from uniformly translating fronts to coherent pattern forming fronts as the
spreading points parameters change when 2 is increased beyond 1/12. Zimmermann [450] has shown
that the equation can arise as a type of amplitude equation in electro-hydrodynamic convection in
liquid crystals. In this interpretation the point 2=1=12 can be viewed as a dynamical Lifshitz point. 43
2.11.2. The Swift–Hohenberg equation
The Swift–Hohenberg equation
9t u = u − (92x + 1)2 u − u3 = ( − 1)u − 292x u − 94x u − u3
(92)
was introduced as a simple model equation for the dynamics just above a supercritical 3nitewavelength instability [389]. 44 In the Swift–Hohenberg equation, the u = 0 state is linearly unstable for ¿ 0; for 0 ¡ 1 the equation gives rise to patterns of wavenumber about 1 whose
dynamics can be analyzed in terms of an amplitude equation. The dispersion relation for small
perturbations about the unstable state is given by (86) with a = − 1 and b = 2.
For ¿ 0 the Swift–Hohenberg equation admits a band of stable periodic states, and this has
made the Swift–Hohenberg equation an attractive model equation for front propagation in the past,
as fronts propagating into the unstable state u = 0 are found to generate one of these regular and
stable periodic patterns behind them—see Fig. 14(a). For this reason the Swift–Hohenberg equation
has played an important role both from a historical and from a practical point of view:
(i) The 3rst numerical study of a pulled pattern forming front was done for this equation and
expression (88) for v∗ was con3rmed in detail [111,421].
(ii) The node conservation that we discussed in the previous subsection on the EFK equation was
3rst observed empirically for this equation by Dee and Langer [111] and this lead these authors
to derive Eq. (91) with a = − 1 and b = 2 for the wave number qstat of the stationary periodic
pro3le generated behind the front. Numerical results were found to be in full accord with this
expression [111,421], indicating, as for the EFK equation, that these front solutions appear to
be of the coherent pattern forming type (56).
(iii) It was the 3rst pattern forming equation for which the transition from pushed to pulled fronts
was studied in detail both numerically and analytically by including either a symmetry breaking
quadratic term u2 or by changing the sign of the cubic term and adding a stabilizing quintic term
[421]. Simulations of a pushed front in the presence of a quadratic term are shown in Fig. 14(b).
43
A Lifshitz point is a point in parameter space where a phase transition from a homogeneous to a modulated state
occurs.
44
It also arises in the context of laser physics [258] or as an amplitude equation near a particular type of co-dimension
2 bifurcation point [369].
84
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Swift–Hohenberg equation
pulled front at ε = 0.4
Swift–Hohenberg equation
2
with term 3φ added and ε = 0.4:pushed front
15
15
10
10
5
5
0
0
0
(a)
30
60
90
120
0
30
60
90
120
(b)
Fig. 14. Space–time plot similar to Fig. 13, but now for the Swift–Hohenberg equation. Total run time is 35, and the time
diDerence between successive lines is 1; the simulations started from a Gaussian initial condition. The plots in this 3gure
are more detailed version of those in the middle column of Fig. 1. (a) A pulled front at = 0:4. (b) Simulation of the
Swift–Hohenberg equation with a symmetry
breaking term bu2 added to the right-hand side. For b ¿ bc () fronts in this
equation are pushed, where bc ( ↓ 0) = 27=38 [421]. The simulation shown is for b = 3 and = 0:4, while bc (0:4) ≈ 1:85
[421]. Note that after a while, a second front develops; at later times this moves into the periodic state generated by the
3rst front. This front generates a stable homogeneous state u = const.
The dynamics of the pushed fronts generated this way was con3rmed to be associated with
the existence of nonlinear front solutions with steepness larger than ∗ , in agreement with the
arguments presented in Section 2.7.
(iv) The Swift–Hohenberg equation is to our knowledge essentially the only pattern forming equation
for which the existence of pattern forming fronts has been proved rigorously and for which the
pulled front selection has been established with some rigor [91–93]. In fact, the 3rst formulation
of what we refer to as coherent pattern forming front solutions appears to have been made for
this equation by Collet and Eckmann [92].
(v) The Swift–Hohenberg equation has recently also been used in extensive numerical tests of the
power law convergence discussed in Section 2.7 of the wavenumber of the pattern generated
behind a pulled front [388]. The results which are reported in Fig. 19 below are found to be
in excellent agreement with the theoretical predictions.
2.11.3. The Cahn–Hilliard equation
The Cahn–Hilliard equation is a simple model equation for the problem of phase separation and
dynamic coarsening [56,192], which we will brieKy review in the context of polymer phase separation
in Section 3.7. The dynamical equation is based on the idea that in the phase separation regime the
coarse grained free energy functional F = d x 12 (9x u)2 − u2 =2 + u4 =4 for the composition 3eld u has
two minima, at u = ±1, and on the idea that this composition 3eld is a “conserved order parameter”.
This means that mass is neither created nor destroyed, but that it can exchange due to diDusion in
response to a free energy gradient. These ingredients lead to the Cahn–Hilliard equation
F
2
= −92x (u − u3 ) − 94x u :
9t u = 9x
(93)
u
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
85
Note that this gradient structure of the equation implies the conservation of composition in the
following form: if we imagine taking a large system of size L with zero Kux boundary conditions,
then it follows immediately that
d L
d x u(x; t) = 0 :
(94)
dt 0
Thus the spatially averaged composition is conserved under the dynamics. 45 The long-time dynamical
behavior implied by (93) is that the system is driven towards a phase consisting of domains where
u is close to one of the two minimal values ±1, separated by domain walls or kinks. In higher
dimensions the dynamics of these domains is driven by the curvature of the domains (“droplets”),
while in one dimension the coarsening is driven by the interaction between the domain walls. This
interaction is exponentially small for large separations, and the coarsening dynamics in one dimension
is therefore quite slow.
Consider now a homogeneous state u = uc = constant. If we linearize the Cahn–Hilliard equation
√
about uc , the resulting equation for u−uc is of the form (85) with a=0 and b=1−3uc2 . For uc ¡ 1= 3
the homogeneous state is thus unstable and one can consider the following
front propagation problem
√
[266,21]: we consider an initial condition where u ≈ uc ¡ 1= 3 everywhere except in a small
localized region near the left edge, where we impose the boundary conditions u = 1, 9x u = 0, and
investigate the front which propagates into the linearly unstable state, whose linear spreading point
values are given by (88) with a = 0 and b = 1 − 3uc2 .
An interesting aspect of the Cahn–Hilliard equation (and of similar equations with conserved
dynamics) is that one can immediately see that even if uniformly translating front solutions of the
type u(x − vt) obeying the boundary conditions exist, they cannot be relevant for the dynamics.
To see this, suppose such a solution exists; for such a solution, we would then have
L
d L
d x u(x − vt) = −v
d x 9x u(x − vt) = −v(uc − 1) = 0 ;
(95)
dt 0
0
in contradiction with (94). A more intuitive way to understand this is as follows: because of mass
conservation, the phase separation in the front region can only occur through the formation of
domains of the two preferred phases. An isolated wall between two domains has no intrinsic motion.
So, the region behind the front is a modulated phase with only a slow coarsening dynamics associated
with the slow motion of domain walls in either direction, and a coherent solution moving with the
front speed is dynamically impossible. 46
The simulations of the Cahn–Hilliard equation of [21,267] and those shown in Fig. 15 con3rm
that fronts in the Cahn–Hilliard equation are pulled in agreement with intuitive notion that the
nonlinearities slow down the growth. Beyond this, not much is known with certainty, as the fronts
in this equation do not appear to have been investigated in any detail. The most likely scenario, it
appears to us, is that the relevant front solutions are incoherent ones, since the coarsening of the
pattern sets in as soon as it is formed. However, it is not excluded that the equation does admit
coherent pattern forming front solutions, and that the state generated by the front is unstable to
45
Moreover, form 2 (93) implies that F is also a Lyapunov function for the dynamics: dF=dt =
d x [9(F=u)=9x] 6 0.
46
Such arguments apply of course more generally to conserved equations. As an example of this, the complex dynamics
of bacterial Kagella traces back partially to the conserved nature of the underlying dynamical equations [97].
−
86
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Cahn–Hilliard equation
Cahn–Hilliard equation
pulled front propagating into unstable state u = 0.4
pulled front propagating into unstable state u = 0.475
20
20
15
15
10
10
5
5
0
0
40
80
120
160
(a)
0
0
40
80
120
160
(b)
Fig. 15. Space–time plot similar to Fig. 13, but now for the Cahn–Hilliard equation, starting from a Gaussian initial
condition. (a) A pulled front propagating into the unstable state u = 0:4. Total run time is 300, and the time diDerence
between successive lines is 6. Note the slow coarsening (domain merging) behind the pro3le at later times. (b) The same
for an homogeneous initial state u = 0:475. The total run time is 500, time diDerence between successive lines is 10. Note
that the front propagates slower than in the case shown in (a), due to the fact that the homogeneous state on the right is
less unstable.
coarsening. Indeed, since during the coarsening process domains can merge, a simple node counting
argument can not be applied to predict the 3nal state pattern. Nevertheless, it does appear from
the plots that such an analysis might give a reasonable estimate of the transient domain size in
the regime where the front propagation speed is rather high. 47 Whether there might possibly be a
transition from coherent to incoherent pattern forming fronts as uc is increased, has to my knowledge
not been investigated.
Thus, while the linear spreading dynamics of the Cahn–Hillard equation is just a special case
of that of the Swift–Hohenberg equation, the dynamical nonlinear behavior of the patterns that the
front gives rise to is very diDerent. This con3rms the picture that a pulled front just propagates as
is dictated in essence by the linear spreading point conditions, while the nonlinear dynamics behind
it just follows in whatever way is allowed or imposed by the dynamics in the nonlinear region.
2.11.4. The Kuramoto–Sivashinsky equation
The Kuramoto–Sivashinsky equation [244,386]
9t u = −92x u − 94x u + u9x u
(96)
has played an important role in the study of nonlinear chaotic dynamics. Even though the equation
admits periodic solutions, it has nearby weakly turbulent states. Even though the latter states may be
transient, they usually dominate the dynamics, as their survival time grows exponentially fast with
the size of the system [381,448]. The scaling properties of these turbulent states, in particular the
47
If so, the picture is somewhat similar to the one shown in Fig. 17(a) for the cubic CGL equation: the 3rst pulled
front generates a state which is unstable, and which hence is invaded by a second front.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Kuramoto–Sivashinsky equation
Kuramoto–Sivashinsky equation
propagation of a pulled front
pushed front in KS with term 0.15u XXXadded
20
20
15
15
10
10
5
5
0
0
0
(a)
87
30
60
90
120
0
30
60
90
120
(b)
Fig. 16. Space–time plot similar to Fig. 13, but now for the Kuramoto–Sivashinsky equation, starting from a Gaussian
initial condition. Total run time is 60, and the time diDerence between successive lines is 1.2. (a) A pulled front propagating
into the unstable state u = 0. The state generated by the front is a weakly chaotic state of the type typically found for
this equation. (b) The same but now for an extension of the Kuramoto–Sivashinsky equation, obtained by adding a term
c93x u to the right-hand side. The results of [79,196] imply that upon increasing c, a transition from pushed to pulled fronts
occurs at c ≈ 0:14. The simulations shown in the plot are for c = 0:15 as in [196], i.e., just above the transition, which
itself is not understood in detail. Note that in this case the transition is tuned by changing a linear term in the equation,
not a nonlinear term! The plots in this 3gure are more detailed versions of those in the right column of Fig. 1, except
that there c = 0:17 in the lower panel.
question whether they are in the so-called KPZ universality class, has also been a subject of intense
research [52,273,367,445].
Clearly, the state u = 0 is unstable due to the negative prefactor of the diDusion-like term, and
the dispersion relation is again of the standard form (86) with b = 1 and a = 0; substitution of these
values into (88) gives the linear spreading parameters for this equation for fronts propagating into
the state u = 0.
Several authors [94] have studied the front propagation problem in this equation both numerically
and analytically. Some representative simulation results are shown in Fig. 16, where starting from a
∗
localized initial condition one obtains two pulled fronts which propagate out with velocity v4th
ord (0; 1)
in opposite directions. Although this has to my knowledge not been investigated explicitly, it is
quite clear from such simulations that these front solutions are inherently incoherent. Whether the
Kuramoto–Sivashinsky admits (unstable) coherent pattern forming front solutions is not known to
me; in any case this issue is probably not very relevant for the dynamics.
Another interesting feature is that when the Kuramoto–Sivashinsky equation is modi3ed by adding
a cubic term c93x u to the right-hand side of (96), then one of the two fronts becomes pushed for a
critical value of c of about 0:14 [79,196]. The pushed fronts appear to be driven by a pulse-type struc∗
ture which propagates faster than v4th
ord , and which leaves behind a series of pulse-type structures
as well. It is an understatement to say that these empirical observations are not very well understood
theoretically. Possibly, in some limit it is more appropriate to think of the pushed dynamics in terms
of the splitting and birth of pulse-type solutions [131,120] than in terms of a propagating front.
88
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.11.5. The cubic complex Ginzburg–Landau equation
The cubic Complex Ginzburg–Landau (CGL) equation
9t A = (1 + ic1 )92x A + A − (1 − ic3 )|A|2 A
(97)
with ¿ 0 is the appropriate amplitude equation just above the onset to 3nite wavelength traveling
wave patterns. The amplitude A is a complex quantity, and describes the slow space and time
modulation of the 3nite wavelength mode that goes unstable [9,105,155,189,316,320,413,426,435].
The phase diagram of the dynamics behavior of this equation as a function of the real parameters
c1 and c3 is extremely rich, and includes chaotic phases as well as coherent dynamics phases
[9,81,105,382,418,419].
An important simple set of solutions of the CGL equation are the periodic or phase winding
solutions
A = aeiqx−i/t ;
(98)
with a a constant. Upon substitution of this expression in to the cubic CGL, it is easy to check that
these solutions form a one-parameter family parametrized by q, as we can express a and / in terms
of q as
√
√
(99)
a2 = − q2 ; / = (c1 + c3 )q2 − c3 ; − ¡ q ¡ :
In an amplitude description, the phase winding solutions represent periodic traveling patterns whose
wavenumber diDers by q from the critical mode at the bifurcation. Because the CGL equation
admits a family of these pattern-type solutions, fronts typically generate a pattern and hence are
pattern forming fronts—although the CGL equation looks super3cially like the F-KPP equation, the
fact that the dynamical 3eld is complex makes its dynamical behavior much more intricate (One
might even say the pulled front propagation mechanism is virtually the only element they have in
common!).
Not all the phase winding solutions are stable. Those near the edge of the band are always linearly
unstable (the so-called Benjamin–Feir instability), while the stability of those near the center of the
band depends strongly on c1 and c3 . In fact, for c1 c3 ¿ 1 none of the phase winding solutions is
stable [105,278,279,315]. Chaotic behavior is typically found in this region of the phase diagram.
The state A = 0 in the cubic CGL (97) is linearly unstable; since the dispersion equation is
quadratic in k, it is easy to obtain the spreading point parameters
∗
∗
vCGL
= 2 (1 + c12 ); ∗ =
;
k
=
c
; D = 1 + c12 :
(100)
1
r
2
1 + c1
1 + c12
CGL equations generally admit an important class of so-called coherent structure solutions of the
form
A(x; t) = e−i/̂t Â(x − vt) ;
(101)
which is a special case of the general form (56) of coherent pattern forming fronts. Coherent
structure solutions of this type—fronts, sources, sinks, pulses [424] and other types of localized
solutions [418,419]—turn out to be the crucial building blocks of the dynamics of the CGL, but
we limit our discussion here exclusively to fronts. The simplicity of the coherent fronts in the CGL
equation lies in the fact that they are of the coherent pattern forming type (56), but that there
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
89
is only one nonzero term in this sum (we can think of the function  as the term ,1 in this
expression). Since  depend only on a single co-moving coordinate - = x − vt,  obeys an ordinary
diDerential equation. The multiplicity of coherent structure solutions (do they come in discrete sets,
one- or two parameter families?) can therefore be studied from the number of stable and unstable
manifolds near the 3xed points that describe the asymptotic behavior of these solutions (so-called
“counting arguments”). For fronts, these counting arguments show [424] that there generically is a
two-parameter family of front solutions which to the right decay exponentially and on the back side
approach a phase winding solution. Since k ∗ is a complex quantity, this implies that there generally
will be a unique coherent front solution (101) with velocity v∗ and which smoothly connects the
∗
∗
linear spreading point behavior eik (x−v t) on the right with a phase winding solution like (98) on
the left.
Since the temporal phase factor is a global (x-independent) factor, one can use an argument
reminiscent of the conservation of nodes argument discussed in Section 2.11.1 to calculate the
wavenumber of the coherent pulled front solution. Indeed, if we compare in the frame moving
with velocity v∗ two points a 3xed distance apart, this phase di>erence is 6xed since Â() is
time-independent in this frame while the prefactor e−i/̂t is common to both points. By equating the
temporal phase winding in this frame at the two points we then get the wavenumber q behind the
front: In the leading edge to the right, /̂ simply needs to be equal to the temporal phase winding in
the moving frame, /̂=!r∗ −v∗ kr∗ =−c1 . Likewise, let us write the asymptotic phase-winding behavior
behind the front for → −∞ as Â() = aeiq ; the amplitude a, wavenumber q and frequency / are
then related by /̂ = (c1 + c3 )q2 − c3 + v∗ q, which is the analog√of Eq. (99)
√ in the moving frame.
Equating the two expressions for /̂ and solving for the root − ¡ q ¡ simply yields for the
wavenumber qsel selected by the coherent pulled front solution [321]
1 + c12 − 1 + c32 √
:
(102)
qsel =
c1 + c 3
Soon after the broader applicability of what we now refer to as the pulled front concept started
to emerge [38,111], Nozaki and Bekki [321] studied the front propagation problem in the CGL
numerically. Their results can be understood simply in combination with the results of the stability
analysis of the phase winding solution qsel :
(i) In large parts of the phase diagram, the 3rst front propagating into the unstable state A = 0
∗
given by (100) and selected
is a coherent pulled front of the type (101), with velocity vCGL
wavenumber qsel given by (102).
(ii) If the selected mode ∼ exp(iqsel x) is stable, a domain with this state remains behind the front. 48
(iii) As Fig. 17 illustrates, if the qsel is unstable due to the Benjamin–Feir instability, the domain
behind the 3rst coherent front is invaded by a second front. This front is again a pulled front
∗ can be calculated explicitly in terms of q
49
[112,321], whose speed vBF
sel and c1 and c3 .
Depending on the parameters c1 and c3 this front can give rise either to a more stable phase
winding solution, or a chaotic state. An example of the latter case is shown in Fig. 17(a).
48
Depending on the initial conditions, this domain can subsequently be invaded by a front connecting to another phase
winding solution.
49
Since the Kuramoto–Sivashinsky equation is the amplitude equation for the Benjamin–Feir instability in the limit when
the instability is weak [105], the second front is essentially a pulled Kuramoto–Sivashinsky front.
90
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
quintic CGL equation
Cubic CGL equation
9
pulled fronts at c1= 1 and c3=3
pulled front at ε = 0.25
24
18
6
12
3
6
0
0
(a)
75
150
225
300
0
0
80
160
240
320
(b)
Fig. 17. (a) Space–time plot of the amplitude A in a simulation of the cubic GL equation (97) with = 1, c1 = 1 and
c3 = 3. Time diDerence between successive lines is 2 and the total simulation time is 80. The plot illustrates how the 3rst
front is a pulled coherent pattern forming front of the type (101), but that the state behind the front is unstable, so that
the domain with the state selected by the 3rst front is invaded by a second front. This second front is also a pulled one.
Note the similarity between the leading part of the second front and the pulled front found in the Kuramoto–Sivashinsky
equation—see Fig. 16(a). This is no accident, as the Kuramoto–Sivashinsky equation is the lowest order amplitude equation
just above the Benjamin–Feir instability [105,189]. (b) Space–time plot of a propagating pulled front in the quintic CGL
equation with = 0:25, c1 = 1, c3 = −1 and c5 = −1. For these parameter values, no coherent nonlinear front solution of
the type (101) exists [424]. The 3rst front is followed by a region with phase slips (points where A vanishes), before
giving rise to a state close to a phase-winding solution. At the later stage, this region is invaded by another front.
Clearly, this regime can only exist if the 3rst coherent front outruns the second one, hence
∗
∗ . In the frame moving with the 3rst front, the state
in the parameter range where vCGL
¿ vBF
generated by the 3rst front is nonlinearly convectively unstable to the Benjamin–Feir instability.
(iv) In the parameter range where the Benjamin–Feir instability becomes nonlinearly absolutely
∗
∗ (which
unstable in the frame moving with the 3rst front, i.e., in the range where vCGL
¡ vBF
∗
is most easily entered by tuning c3 as vCGL does not depend on c3 ) the coherent pulled front
solution is irrelevant. Instead, one observes a pulled incoherent front: 50 the chaotic behavior
sets in right at the front region front. The dynamical behavior in this regime is similar to that
shown in Fig. 19(a) below for the pulled chaotic behavior of the quintic CGL.
In conclusion, the cubic CGL equation admits a two-parameter family of coherent front solutions.
This family includes the pulled front solution, for which the selected wavenumber can be determined
∗ ¡ v∗
explicitly. The dynamics behind the 3rst front depends on the stability of this state. When vBF
CGL
the 3rst front is a coherent pulled front, while in the opposite regime an incoherent pulled front
emerges.
50
Actually, it is conceivable that in some parameter ranges the Benjamin–Feir instability of the nonlinear region is so
strong, that the very 3rst front becomes a pushed incoherent front. To my knowledge this issue has not been studied
systematically.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
91
2.11.6. The quintic complex Ginzburg–Landau equation
Unlike the cubic CGL equation, which arises as the lowest order amplitude equation near a
supercritical (forward) bifurcation to traveling wave patterns, the quintic CGL equation
9t A = (1 + ic1 )92x A + A + (1 + ic3 )|A|2 A − (1 − ic5 )|A|4 A
(103)
is a model equation for the case in which such a bifurcation is subcritical (backward). Because
of the diDerent sign of the cubic term, this term now enhances the growth of amplitude, while
saturation is only caused by the quintic term. Indeed, the destabilizing cubic term implies that the
only stable phase winding solutions have a 3nite amplitude for any . So an expansion based on
assuming that the amplitude is small is not really justi3ed. In other words, the quintic CGL is not
a consistent lowest order amplitude equation. Nevertheless, the equation has played an important
role in identifying the main coherent structures and dynamical regimes near a subcritical bifurcation.
Note that the terms linear in A are the same in the quintic and cubic CGL, so the linear spreading
point expressions (100) apply to the quintic CGL equation for ¿ 0 as well.
In the subcritical range, ¡ 0, the state A=0 is linearly stable, and hence a small perturbation does
not spread with a 3nite speed: instead it will die out. However, nonlinear states, in particular phase
winding solutions, can perfectly well be stable. In analogy with thermodynamic systems, where one
is used to interface-type solutions between two stable phases separated by a 3rst order transition, one
expects there to be front solutions in the subcritical range ¡ 0. When is increased towards small
but positive values, these front solutions would then be expected to remain√dynamically relevant as
∗
they have nonzero speed larger than the linear spreading speed vCGL
∼ . In other words, one
would naively expect that for small fronts in the quintic CGL are pushed.
The remarkable feature of the quintic CGL is that the front solutions corresponding to the pushed
fronts can be obtained analytically [423,424] by a generalization of the “reduction of order method”
reviewed brieKy in the example at the end of Section 2.7.1: When Â(-) with - = x − v† t is written
in the form  = a(-)ei(-) and the Ansatz
da
d
(104)
= e1 (a2 − a2N );
= qN + e0 (a2 − a2N ) ;
ddis substituted into the equations, one arrives at algebraic equations for the coeRcients e0 and e1 and
the wavenumber qN and amplitude aN of the phase winding solution behind this front solution. These
coherent pattern forming pushed front solutions, which are unique for a 3xed set of parameters, only
exist in parts of the ; c1 ; c3 ; c5 parameter space: EDectively they exist only in a band around the
line c3 = −c5 where the nonlinear dispersion terms almost cancel and especially in the subcritical
range ¡ 0.
The main 3ndings of a detailed study of the existence of these solutions and of their competition
with pulled front solutions are:
(i) Just like we discussed for the pulled front solutions of the cubic CGL equation, the phase
winding solution with wavenumber qN behind the pushed front can be unstable to the Benjamin–
∗ is smaller than v† , then the size
Feir instability. If it is, and if the propagation velocity vBF
∗ )t in time. If v∗ ¿ v† , however, an
of the domain behind the 3rst front grows as (v† − vBF
BF
incoherent pattern forming pushed front results (see Figs. 13 and 14 of [424]).
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(ii) Even though the quintic Ginzburg–Landau equation always admits phase winding solutions in a
subcritical range c ¡ ¡ 0, outside a band where c3 ≈ −c5 there are no pushed front solutions.
This means that the “thermodynamic” intuition formulated above, according to which one always
expects a pushed to pulled transition at some 6nite positive value of , does not apply to the
quintic CGL equation (in line with the fact that there is no standard Lyapunov function for this
equation). In other words, in some parameter ranges, the dynamically relevant fronts for any
positive are pulled fronts! This is quite remarkable, as it means that in the parameter ranges
where this happens, from the point of view of the front propagation problem the dynamical
behavior is more like that found near a supercritical bifurcation. 51 This also illustrates that
while for the F-KPP equation the nonlinear behavior of the growth function f() determines
whether fronts are pushed or pulled, the distinction between the two regimes is generally more
subtle—in the quintic CGL it is determined by the strength of the nonlinear dispersion, in the
extension of the Kuramoto–Sivashinsky equation it is tuned by a symmetry-breaking linear third
order derivative term.
(iii) In equations where the relevant front solutions are uniformly translating ones, the pushed solutions normally bifurcate oD the pulled ones continuously. In the quintic CGL equation, this
is generically not the case for pattern forming fronts—see the discussion in Section 2.8. As a
result, the selected wavenumber can jump as one goes from a coherent pulled front solution
to a coherent pushed front solution (see Fig. 1(c) of [423] or 23 of [105]). Thus, the coherent pattern forming front solutions found in the quintic CGL provide a counterexample to the
“structural stability” postulate [333,334] that pulled front solutions are smoothly connected to
the pushed front solutions, as one tunes one of the parameters in the dynamical equation.
(iv) As we discussed, for the cubic CGL equation, so-called “counting arguments” for the multiplicity of front solutions indicate that there generally is a two-parameter family of front solutions.
Although this has not been checked explicitly, we believe that for any value of the parameters
there is indeed a unique pulled front solution, in line with the fact that (102) 3xes qsel uniquely.
For pulled fronts in the quintic CGL equation, a new feature is encountered: an extension of
the earlier analysis to the quintic case shows that in some parameter ranges the equations for
qsel behind the front do not admit a solution. This implies that in these parameter ranges no
coherent front solutions (101) can exist [424]. Fig. 17(b) shows an example of the type of
dynamical behavior that is found in this regime: in the front region the dynamics is very incoherent and associated with the occurrence of “phase slips” (points in space-time where A = 0).
The fact that the incoherent front region is followed by a zone where A is close to a stable
phase-winding solution illustrates that the incoherent front dynamics in this case is not due to
an instability of the selected state, but to the absence of a coherent pulled front solution.
(v) Because of the subcritical nature of the transition, the quintic CGL equation also has parameter
ranges at ¡ 0 where a chaotic phase is statistically stable. If one puts such a chaotic phase
next to a domain of the stable A = 0 phase, the front between them can move. Upon increasing
one can then observe a pushed to pulled transition of an incoherent front—see Fig. 18.
(vi) As we discussed in Section 2.9, the universal 1=t power law relaxation even applies to the coherent and incoherent pattern forming fronts. This was checked numerically for the simulations
51
For ¡ 0, a new class of retracting fronts (v ¡ 0) was recently found [103]; these solutions determine much of the
dynamical behavior in the subcritical regime, especially outside the band where the pushed front solutions exist.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
quintic CGL equation
quintic CGL equation
subcritical range ε = -0.005
21
93
pulled front at ε = 0.03
21
14
14
7
7
0
0
0
120
240
360
(a)
0
480
120
240
360
480
(b)
Fig. 18. Space–time plot of A illustrating the pushed to pulled transition for incoherently propagating fronts in the quintic
CGL equation (103) [119]. In each plot, the total simulation time is 100, and the time diDerence between successive lines
is 2. (a) Simulation for c1 = −2:2, c3 = −0:5, c5 = −2 in the weakly subcritical range = −0:005. According to Fig. 15
of [424], in this range the incoherent domain in the center slowly spreads, corresponding with a pushed incoherent front
with velocity ≈ 0:15, although this may be a 3nite-size eDect. (b) The same for = 0:03.
(a)
(b)
(c)
Fig. 19. (a) Space–time plot of the amplitude |A| of a pulled incoherent front in the quintic CGL at = 0:25, c1 = 1,
c3 = c5 = −3. Note that in the leading edge, where |A| is small, the dynamics is actually quite coherent, in agreement
with the fact that the linear spreading dynamics is always coherent. The front only becomes incoherent once the nonlinear
behavior sets in. (b) Scaling plot of the scaled velocity relaxation [v(t) − v∗ ]T=C1 versus T=t for simulations of coherent
fronts in the Swift–Hohenberg equation and the quintic CGL simulations shown in (a). Here T is a properly de3ned time
scale introduced so that both equations fall on the same asymptotic curve, and C1 is an appropriate combination of the
linear spreading
From left to right, the 3rst three lines are Swift–Hohenberg data with = 5, at
√ point
√ parameters [388].
√
heights u = ; 0:01 and 0:0001 , and the next three lines show the CGL data traced at amplitudes 0:002, 0:0002 and
0:00002. The solid line is the universal asymptote predicted by the analytic theory. (c) The same for the appropriately
scaled 5̇(t) relaxation (84). From [388].
of the incoherent fronts of the CGL equation shown in Fig. 19(a). The other panels con3rm that
after scaling the velocity and time with the predicted values, both the velocity [panel (b)] and
wavenumber [panel (c)] relaxation data of the incoherent quintic CGL fronts and the coherent
fronts in the Swift–Hohenberg equation are in accord with the universal predictions.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
2.12. Epilogue
The insight we have attempted to bring across in Section 2 is that the special nature and generality
of front propagation into unstable states lies in the fact that there is a well-de3ned linear spreading
speed v∗ associated with any linear instability. The linear spreading point is the organizing center
for all subsequent developments: We can make a large number of general predictions because the
linear dynamics puts a strong constraint on any possible nonlinear dynamics. On the one hand, it
forces us to focus on the question “how can a front faster than v∗ emerge?” and this naturally
leads to the concepts of pulled and pushed front. On the other hand it shows that the approach
to the asymptotic velocity and state of a pulled front is governed by an equation which in lowest
order is of diDusion-type, and this allows one to derive the universal asymptotic relaxation of a
pulled front.
For uniformly translating fronts and coherent pattern forming fronts, the regime of pushed front
propagation is essentially governed by the existence of special solutions which we can think of as
“strongly heteroclinic orbits”. For incoherent pushed fronts we are at present unable to formulate the
pushed front propagation mechanism sharply, although intuitively we expect it to be very similar to
that of the other two types of fronts.
The various examples in the previous section illustrate that a pulled front essentially does what
it should do, namely propagate with asymptotic speed and shape dictated by the linear spreading
point, while the rest of the front just follows in whatever way is allowed by the global nonlinear
properties of the dynamical equation: (i) a bifurcation in the structure of the linear spreading point
is responsible for the transition to coherent pattern forming fronts in the EFK equation; (ii) the
quintic CGL equation illustrates that the limited range of existence of coherent pattern forming
fronts can lead to a transition from coherent pulled fronts to incoherent pulled fronts; (iii) if the
state generated by a front in the cubic or quintic CGL equation is unstable, the 3rst front remains a
coherent front if the spreading speed of the secondary instability is less than v∗ of the 3rst front, while
the 3rst front becomes incoherent if the pulled speed of the secondary front is larger than v∗ of the
3rst front.
The diRculty in making generic statements about pushed fronts is that apart from the fact that their
convergence to the asymptotic speed v† is exponential, there are few generic features. Because of the
strong heteroclinicity requirements (60) or (76) that the slowest decaying exponential mode is absent,
pushed front solutions are technically determined by an analysis which has all the properties of a
nonlinear eigenvalue problem for an isolated solution. Because of this structure, the relaxation towards
the asymptotic pushed front speed and shape is generally exponential, and singular perturbation theory
can be applied to pushed fronts (Section 5). Moreover, because of this structure, whether a pushed
front solution exists depends on the full global and hence nonlinear properties of the equations. As
a result, virtually every example is special, and there are few nontrivial examples which one has
been able to obtain analytically. The quintic CGL equation is a remarkable exception in this regard.
As we discussed, with few exceptions, most mathematical investigations of front propagation into
unstable states have focused on uniformly translating fronts and in particular on systems of equations
which are closely related to the F-KPP equation. It appears to us that many of the ingredients of
the theoretical framework that we have discussed here are ready for more rigorous analysis: rigorous
developments that I expect are around the corner are (i) rigorous bounds for the Green’s function
of whole classes of linearized equations and, in relation with this, rigorous proofs of the pulled
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
95
front mechanism of new classes of equations, (ii) proofs of the universal relaxation of pulled fronts,
(iii) the development of a mathematical framework for the pulled to pushed transition of coherent
pattern forming fronts, and (iv) extension of the approach of Sandstede and Scheel [374,375] to
analyze the interaction between front dynamics and the dynamics of the state behind the front, or
to classify defects.
3. Experimental and theoretical examples of front propagation into unstable states
In this section we discuss a number of examples of front propagation into unstable states. We 3rst
review experiments on fronts or in which front propagation plays an important role, and then move
on to discuss examples of fronts in theoretical models. The examples concern mostly fronts in one
dimension. The diRculty of analyzing patterns whose dynamics is governed by the propagation of
pulled fronts is discussed later in Section 5.
At 3rst sight, one might well wonder why front propagation into unstable states is a relevant
problem anyway. After all, it seems counterintuitive that a physical system would naturally end up
in an unstable steady state and wait for a nice front to develop! On closer inspection, however, there
are many reasons why front propagation into unstable states is not an esoteric problem:
(i) A natural way in which a system can stay in a self-sustained unstable state is when an overall
Kow or motion makes the instability convective (see Sections 2.5 and 2.10). This happens in
particular in many Kuid dynamic systems. Likewise, in plasma physics the strong asymmetry between the mobility of electrons and ions usually makes plasma instabilities convective.
Examples where these considerations apply are discussed in Sections 3.5, 3.9, 3.10, 3.11, 3.12,
and 3.17.
(ii) Sometimes a natural way to probe a system is to quench it into an unstable state with a laser
pulse, ramping or Kipping a voltage, Kipping a plate, dropping a temperature, scraping a skin
tissue, etc. In all these cases, the relevant question is whether we can change the system on a
timescale shorter than the natural time scale of the evolution of the state into which we quench
it. Examples are encountered in Sections 3.1, 3.2, 3.3, 3.5, 3.6, 3.7, 3.14, 3.15, and 3.20.
(iii) We sometimes encounter intrinsically chaotic systems where the dynamics itself continuously
generates states which are transient because they themselves are unstable. We discuss a clear
example of this in Section 3.4.
(iv) In thermodynamic systems with 3rst order transitions, interfaces play a predominant role because
the bulk nucleation rate is exponentially small near the transition, while interface velocities
usually vary linearly with the distance from the transition. Although Kuctuations near a second
order thermodynamic transition are usually too large to allow clear fronts to develop, this is not
necessarily true for polymer systems (Section 3.7). Moreover, near supercritical (“second order
like”) bifurcations in pattern forming systems, where Kuctuation eDects are usually small [105],
a similar argument holds as well: as we shall discuss more explicitly in the next subsection, if
is the dimensionless control
√ parameter measuring the distance from the instability threshold,
the front speed scales as while the growth rate of bulk modes scales as . Hence, front
propagation in principle always dominates for suRciently small [161]. Of course, the range
over which fronts do dominate the dynamics depends on the prefactors, i.e. on the time and
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
length scales of the problem under investigation. Such considerations play a role in the examples
of Sections 3.1, 3.2, 3.3, 3.5, 3.7, 3.8, and 3.14.
(v) Front propagation into an unstable state sometimes emerges theoretically from an unexpected
angle through mapping a seemingly unrelated problem onto a front propagation problem—see
Sections 3.21, 3.23, 3.25, 3.26.1 and 3.26.5.
3.1. Fronts in Taylor–Couette and Rayleigh–BIenard experiments
Soon after Dee and Langer [111] and Ben-Jacob and co-workers [38] drew the attention of the
physics community to the special simplicity of what we now refer to as pulled fronts, two experiments
on pattern forming systems were done that are still very illuminating: One by Ahlers and Cannell
[4] on vortex fronts in a Taylor–Couette cell, and one by Fineberg and Steinberg [161] on front
propagation in a Rayleigh–BIenard cell.
A Taylor–Couette cell consists of two concentric cylinders which can rotate independently. The
gap between the two cylinders is 3lled with a normal Newtonian Kuid. When the inner and outer
cylinder rotate slowly, the Kow between the two cylinders is laminar and in the azimuthal direction.
At higher rotation rates, one 3nds an amazing number of Kuid dynamics instabilities when both the
inner and the outer cylinder are rotating with suRciently high frequencies [105,125]. The experiments
that are of interest to us probed fronts that lead to the 3rst nontrivial patterned state that emerges if
the rotation rate of the inner cylinder is increased while the outer cylinder does not rotate [4]. At
some critical rotation rate, there is a bifurcation to a state with so-called Taylor vortices: a pattern
which is periodic in the direction along the axis of the cylinders emerges. This Kow is referred to as
Taylor vortex Kow, since the velocity in the r; z plane has the appearance of alternating clockwise
and counterclockwise rotating vortices that span the whole gap between the two cylinders.
A Rayleigh–BIenard cell consists of a Kuid sandwiched between two parallel plates at a 3xed
but diDerent temperature. When the bottom plate is hotter than the top plate, the Kuid density is
largest at the top. For small enough temperature diDerences, there is no convection—there is just
heat conduction through a quiescent Kuid. However, when the temperature diDerence and hence
the density diDerence becomes larger than some threshold value, this simple state is unstable and
convection sets in spontaneously [105,155,189,282]. The convection patterns that emerge just above
threshold are often referred to as rolls, but if we look at the velocity pattern in a vertical cross
section, they look like the vortex-type patterns in the r; z cross-section of the Taylor–Couette cell.
The transition to Taylor vortex patterns and to convection patterns in the Rayleigh–BIenard case
are both examples of forward (supercritical) bifurcations to a stationary 3nite wavelength patterns.
Just above onset, the patterns can be described by a so-called amplitude expansion by writing the
physical 3elds in terms of the complex amplitude as [105,189,193,316,320]
physical 3elds ˙ Aeikc x + A∗ e−ikc x :
(105)
For simplicity, we restrict the analysis to one-dimensional patterns as this is the relevant case for the
front experiments; kc is the wavenumber of the pattern at onset, and x is the coordinate along which
the patterns develop. The amplitude expansion simply implements the observation that just above
onset the amplitude of the patterns is small, and that the amplitude A varies on slow space and
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
97
time scales. In lowest order, A then obeys the so-called real Ginzburg–Landau amplitude equation 52
[105,155,189,316,320,413,435]
$0 9t A = A + 20 92x A − g|A|2 A :
(106)
Here is the dimensionless control parameter, which for ¿ 0 is a measure for how far one is above
the instability threshold. The time scale $0 and length scale 0 depend on the particular system, and
can be calculated explicitly from the linear dispersion relation of the system under study. This
has been done both for the transition to Taylor vortices and for the transition to Rayleigh–BIenard
convection patterns. The nonlinear parameter g, which follows from the amplitude expansion, is also
known for both systems, but it, of course, does not play a role for the pulled fronts of interest here.
The real amplitude or Ginzburg–Landau equation should in general not be thought of as just a
simple straightforward generalization of the F-KPP equation to a complex 3eld—while the F-KPP
equation only allows for stable homogeneous solutions u = ±1, the amplitude equation admits stable
phase winding solutions of the form A = aeiqx with a and q constant [105,155,189,320,435]. As
(105) illustrates, these describe physical patterns with a wavenumber kc + q diDerent from the
critical wavenumber. The interaction and competition between the various modes in general makes
the dynamics of the amplitude A much more interesting and complicated than that of the F-KPP
equation. Nevertheless, since the dynamics of pulled fronts is dominated by the linear terms in A,
and since these are the same for the two equations, both the asymptotic pulled front solutions and
the convergence to them are the same for the two equations.
The attractive feature of the experiments is that because 0 and $0 are known explicitly, they oDer
the possibility to study front propagation quantitatively. Indeed, for the amplitude equation in form
(106) we get for the pulled front speed
0 √
∗
vampeq
=2
:
(107)
$0
This result explicitly con3rms the assertion made already in point (iv)
√ at the beginning of Section
3 that near a supercritical bifurcation, the front speed increases as , while the growth rate of
bulk modes according to (106) increases linearly in . As stressed by Fineberg and Steinberg [161],
front propagation into unstable states can therefore generally be observed at small enough , i.e. just
above the onset of the transition.
In order to compare the measured front velocities to the F-KPP equation in the standard form in
which the prefactors of the linear terms are equal to 1, the experimental results have been reported
in terms of the scaled velocity
$0
ṽ = v √ :
(108)
0 ∗
if the initial condition is suRciently localized,
Of course, front speeds should only converge to vampeq
as explained in Section 2.7.6. In both experiments, the system was initially below the instability
threshold, corresponding to ¡ 0 in the amplitude equation (106), and then brought above threshold
( ¿ 0) at time t = 0. In the Taylor–Couette experiment, even below threshold, when the bulk of
52
The equation—used as the amplitude equation (106) in the discussion of fronts in Rayleigh–BIenard and Taylor–
Couette experiments in Section 3.1 below—is called the real Ginzburg–Landau equation because the prefactors of all the
terms are real. The amplitude equation for traveling wave patterns is the CGL equation; as discussed in Section 2.11.5,
in principle all its terms have complex prefactors [105,189,435].
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 20. (a) Velocity data of fronts in the Taylor vortex experiments of Ahlers and Cannell
[4]. Data are plotted on a
√
log–log scale as a function of . The solid lines have slope 12 and hence illustrate the scaling of the velocity. The
lower line shows that the data are consistent with a scaled velocity ṽ ≈ 1:1 instead of the asymptotic values ṽas = 2.
(b) Experimental data of Fineberg and Steinberg [161] on fronts in the Rayleigh–BIenard system on a log-linear scale. The
solid line is the theoretical prediction (107). The inset shows the scaled velocity ṽ which is consistent with the theoretical
asymptotic value 2, within the error bars.
the cell is in the stable laminar Kow state, there are so-called Ekman vortices at the ends of the
cylinders. These play the role of localized nonzero initial conditions for the amplitude A when the
system is brought above threshold at time t = 0. For the Rayleigh–BIenard experiment, Fineberg and
Steinberg used a long cell, so as to get one-dimensional fronts, and they turned on an extra heater
at the cell ends as well, when they brought the system to ¿ 0. This made the convection patterns
nucleate at the ends and then propagate into the bulk of the cell.
Fig. 20(a) shows the velocity of the Taylor vortex fronts as they propagate into the Taylor–Couette
cell over a distance of up to√10 to 15 vortex diameters [4], plotted on a log-log scale. As the
two solid lines indicate, the -scaling of the vortex velocity is con3rmed, but the prefactor is
only about 55% of the predicted asymptotic values ṽ∗ampeq = 2: the data are consistent with ṽ ≈ 1:1
instead of 2.
The fact that the measured front velocity was signi3cantly below the asymptotic value in this 1983
Taylor–Couette experiment made many researchers wonder whether there was something crucially
wrong, 53 and stimulated Fineberg and Steinberg a few years later to perform experiments on fronts
in Rayleigh–BIenard convection. Actual space–time plots of their convection front pro3les as they
propagate to the right along their cell, are shown in Fig. 21(a). These traces illustrate how these fronts
(as well as those in the Taylor–Couette cell) are examples of true pattern forming fronts propagating
53
It was realized that if the measured velocity had been too high, a possible candidate for the discrepancy could have
been that the initial convection pro3le associated with the Ekman end vortices
was not suRciently localized, i.e., that at
√
∗
t = 0 the amplitude did not fall oD to the right faster than e− x = e− x=0 . However, as we discussed in Section 2.6,
asymptotic front speeds below v∗ are impossible.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
99
(b)
Fig. 21. (a) Space–time plot of a propagating front in the Rayleigh–BIenard experiment of Fineberg and Steinberg [161]
for = 0:012, made using the shadowgraph technique. The time between successive traces is 0.42 tv , where tv is the
vertical diDusion time in the experiments, and distances are measured in units of the cell height d. (b) A space-time plot
obtained from a numerical simulation of the Swift–Hohenberg equation at the same value = 0:012 and length and time
scales adjusted to correspond to those in the experiments of (a). After Kockelkoren et al. [233].
into an unstable state, like those observed in simulations of the Swift–Hohenberg equation (92), see
the central panel of Fig. 1, and Figs. 14(b) and 21(b).
The data of Fig. 20(b) for the scaled front velocity ṽ, calculated from the actual data by using
the theoretical values for 0 and $0 in (108) were found to be consistent to within a few percent
with the asymptotic value ṽas = 2. At the time, this was considered to be an important experimental
con3rmation of the quantitative prediction of the pulled front speed, and of the idea that the concept
of a pulled front applies more generally to pattern forming systems. But, as we shall see, there is a
new twist to the story.
The reason for the apparent discrepancy between theory and experiment on the Taylor vortex fronts
remained unresolved till Niklas et al. [319] performed an explicit numerical simulation of such fronts
which showed that the front speed had not yet relaxed to its asymptotic value, in contrast to what
was found in numerical studies of the propagating fronts in the Rayleigh–BIenard cell [271]. Indeed,
over the range of times probed experimentally, the transient speeds they found for the Taylor vortex
fronts were quite consistent with the experimental data. Instead of showing these results, we will
approach this issue by extrapolating back from the asymptotic formula (78) as this allows us to
discuss both experiments in a uni3ed way.
When (78) is applied to the amplitude equation, one 3nds for the long time rate of approach to
the asymptotic speed ṽ∗ = 2
√
3 3
+
+ ··· :
(109)
ṽ(t) = 2 −
2t=$0 2(t=$0 )3=2
This formula shows that in any experiments on fronts just above the onset of a 3nite-wavelength supercritical bifurcation, one can a priori only hope to observe front speeds very close to the asymptotic
value at times
ttco ≡ $0 = :
(110)
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Note that we have de3ned the crossover time tco as the time at which the third correction term
is equal in magnitude to the 3rst correction term in (109). Although this asymptotic formula is
only valid in the long-time regime when the third term is much smaller than the second one, and
although for times of O(tco ) one is in the non-universal crossover regime, the important message is
that the velocity always approaches the asymptotic value from below and that transient eDects are
very signi3cant, even for times of order several tco . For example, even at times of O(3tco ) when
the formula becomes reasonably accurate [144], the velocity is still some 8% below its asymptotic
value, and at tco both corrections terms, though of opposite sign, are about 24% of the asymptotic
value in magnitude.
When an actual experiment is suddenly brought into the unstable parameter range ¿ 0, bulk
modes also start to grow at a rate proportional to . Since one can only study fronts properly as
long as bulk Kuctuations have not grown enough that bulk “nucleation” starts to become visible, in
practice experiments like those we consider here can only be done up to a time which scales as −1
[4,161]. In other words, experiments can in practice only be done up to a time which is a 6nite
multiple of tco !
In the Taylor vortex front experiments, most measurements for = 0:02 were done in the time
interval of 1–2 tco . It is therefore not surprising that transient eDects did play an important role
in these experiments [319,422] and that the measured velocity was signi3cantly below the asymptotic value. However, from our present perspective, it is important to turn the question around and
ask whether the fronts observed in the Rayleigh–BIenard experiments were indeed propagating with
approximately the asymptotic speed, and if so, why so [233].
As it turns out, in the experiments of Fineberg and Steinberg, growth of bulk modes also limited
the observation of fronts to times of order tco [161]. Kockelkoren et al. [233] therefore reanalyzed
the experiment from this perspective, and concluded that most likely the actual measured front speed
was some 15% below the asymptotic one. In their interpretation, the theoretical value for 0 used
in converting the observed front speeds to dimensionless front speeds ṽ may have been about 15%
smaller than the actual one. 54 If so, the velocity had been overestimated by about this amount in
the interpretation of the experiments, thus hiding a possible transient eDect.
We stress that only new experiments can settle whether this interpretation is the correct one.
However, there are also indications in the actual traces of the fronts in Fig. 21(a) that the velocity
in this run was slowly increasing in time. The solid line in Fig. 22(a) shows the experimental
velocity as a function of time, extracted by measuring successive front positions with a ruler and
using a value of 0 which is 15% larger than the theoretical value [233]. For comparison, a plot
of the velocity as function of time in a simulation of the real Ginzburg–Landau equation (106) is
shown with a dashed line.
One important point has not been included in the theoretical analysis so far. When growth of bulk
modes in practice becomes important for times of order tco , then at the latest times the growth of the
bulk Kuctuations may become important. This will have the eDect of increasing the velocity above
the one studied analytically and numerically with localized initial conditions and in the absence of
Kuctuations. With this caveat, we nevertheless tentatively conclude that both experiments on fronts
54
Indications that this may be the case come from fact that the observed wavelength at onset was also about 13% bigger
than the theoretical one. See [233] for further discussion of this.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
101
(b)
Fig. 22. (a) Full line: scaled velocity versus time in the Rayleigh–BIenard front [161] whose space-time plot is shown in
Fig. 21(a). The velocity is obtained by interpolating the maxima of the traces. As explained in the text, a value of 0
which is 15% bigger than used by Fineberg and Steinberg to convert the data. A diDerent value would not change the
shape of the curve, only the absolute numbers on the vertical axis. The dashed line shows the measured velocity in a
∗
simulation of Eq. (106) with an exponential initial condition A(x; 0) ∼ e−1:2 x and the dotted line the asymptotic result
(109) with the t −3=2 term excluded. Note that the curves are not 3tted, as only the absolute vertical scale is aDected by our
choice of 0 . (b) Open circles: data-points for the wavenumber selected by the front in the experiments of Fineberg and
Steinberg. Dashed curve: q∗ =qc for the Swift–Hohenberg equation with parameters corresponding to the Rayleigh–BIenard
experiment. The fact that the data follow the line q∗ =qc for small , indicates that just above threshold the experiments
probe the leading edge of the front. After Kockelkoren et al. [233].
in pattern forming systems illustrate that such fronts are prone to transient behavior associated with
the slow power law relaxation of pulled fronts from below to v∗ .
As a 3nal note, we point out that both experiments also addressed the wavelength selection by
the front. Wavelength selection goes beyond the lowest order amplitude equation (106), since the
real F-KPP-type fronts in this equation
simply yield a wavenumber equal to the critical wavenumber
√
kc throughout the front, to order . For the changes in order , one needs to start from the linear
dispersion relation of the problem, correct to order . In the original paper on the Rayleigh–BIenard
experiment, the data were compared to the results for the selected wavelength in the Swift–Hohenberg
equation (92). Strong deviations were found in this case. The most √
likely explanation for this is the
following [233]: for small , the full front whose width scales as 1= is too wide to even 3t in the
experimental cell; hence one then cannot measure the selected wavenumber behind a fully developed
pulled front, but instead only an eDective wavenumber in the leading edge of the front which is
close to the local wavenumber kr∗ given in (88). Fig. 22(b) shows that indeed the Rayleigh–BIenard
data for small are consistent with this.
More recently, the problem of fronts in the Rayleigh–BIenard and Taylor–Couette system has been
analyzed in great detail by L]ucke and co-workers [68–70,210,269–271,349,361], both in the case
discussed here and in the presence of a throughKow (see Section 3.11). In these approaches, the front
predictions based on the lowest order amplitude equation are compared in detail with those for pulled
fronts obtained by determining the linear spreading point of the full Navier–Stokes equations for this
system, and with full numerical simulations. We refer to these papers for a detailed discussion of
how the full result starts to diDer from the lowest order amplitude result as increases. In particular,
the eDect that the nodes of the 3elds drift in the front region before coming to rest behind the front
is a characteristic eDect that goes beyond the amplitude equation (the Swift–Hohenberg equation also
shows this, of course).
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 23. (a) Photograph of a well-developed array of pendant drops at the end of the experiments of Fermigier, Limat,
Wesfreid et al. [158,159,265]. The drops are seen from above through the glass plate across which the 3lm was spread
before the system was made unstable by Kipping the plate. From [265]. (b) Two-dimensional patterns observed in the
Rayleigh–Taylor instability of a thin layer in the same experiments [159,265]. Note that a “roll” or “stripe” type pattern
generally appears near the boundary of the regions. The two structures labeled A and A6 were initiated by small dust
particles on the interface. When they grow, they usually develop into a six-3eld symmetric hexagonal pattern like in H.
The pattern near R was initiated by the thickness gradient close to the edge of the Kuid layer.
3.2. The propagating Rayleigh–Taylor instability in thin 6lms
The name Rayleigh–Taylor instability refers to gravitational instabilities of the interface between
two Kuids or of strati3ed Kuids [78]. A simple example is when one has two immiscible Kuids
separated by a planar interface, in which the heavier Kuid is on top of the lighter one. More
generally the instability arises in density-strati3ed Kuids, when the acceleration is directed from the
heavier to the lighter Kuid. The instability is important in a variety of practical situations, ranging
from explosives to the spreading of paint or coating on a solid surface.
About a decade ago, Fermigier, Limat, Wesfreid and co-workers [158,159,265] performed a series
of experiments on the Rayleigh–Taylor instability of thin 3lms. They studied pattern selection by
3rst spreading thin 3lms of silicon oil on a Kat solid plate, and then making the system unstable
by Kipping this plate upside down. For 3lms of suitable thickness, the instability develops slowly
enough that it can easily be studied. The photo of Fig. 23(a) illustrates that the typical hexagonal
patterns that one often observes after the instability is well-developed. The experiments illustrate
several basic issues in pattern formation, 55 but we will focus here on the propagating fronts. Right
55
For example, the generic appearance of hexagonal patterns is associated with the fact that the thin 3lm equations
(112) below for the deviation - of the 3lm thickness from the uniform value, are not invariant under a change of sign
of - (see [51] and references therein). In the Rayleigh–BIenard instability hexagonal patterns generically arise for the
same reason very close to threshold, since very small non-Boussinesq eDects break the symmetry that the Boussinesq
equations happen to exhibit. Furthermore, the fact that the pattern looks stripe-like near the edge, even though the pattern
is hexagonal in the bulk—see Fig. 23(b)—is a generic phenomenon (also known in convection, see e.g. [51]) which can
be understood simply from symmetry considerations [350]: When one considers a front or domain wall solution in the
amplitude equations for the three modes necessary to describe a hexagonal pattern, symmetry considerations dictate that
the prefactor of the spatial second derivative term of the modes whose wavevector is normal to the front is a factor two
larger than those of the other two modes whose wavevector makes an angle of 60◦ with the front normal. This means
that the “coherence length” or correlation length of this 3rst mode is larger than that of the other two modes, and that
the pattern in the front region looks stripe-like [350] (see also [106,126]).
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
103
Fig. 24. Time evolution of the thickness pro3le as a Rayleigh–Taylor front propagates into an unstable state in the
experiments of Fermigier, Limat, Wesfreid and co-workers [158,265]. The pro3les have been shifted up for clarity and
are separated by time intervals of 30 s. x designates the coordinate perpendicular to the stripe-type pattern in the front
region.
after the plate has been Kipped, a 3lm of uniform thickness is linearly unstable. In many cases,
the instability 3rst developed near the boundary of the sample and then propagated into the bulk
of the sample. Fig. 24 shows experimental traces of such a front as it propagates into the unstable
state [265].
The equations that describe the evolution of a thin liquid 3lm are well known: the separation
of length scales allows one to simply integrate out the coordinates perpendicular to the 3lm layer
(“lubrication approximation”) [324]. If we denote for the experimental case of the free 3lm Kow
the 3lm height h as the thickness of the 3lm measured from the plane, the dynamical equation for
h becomes [324,158]
1
9t h + ∇⊥ · [h3 ∇⊥ (<gh + 2∇2⊥ h)] = 0 ;
(111)
3;
where ; is the viscosity, < the density of the liquid, g the gravity, 2 the surface tension, and ∇⊥
the gradient operator in the direction along the surface. To study the stability of a 3lm of uniform
thickness h0 , we note that if we substitute h = h0 + - into (111), the equation can be writted in the
form
9t - +
h30
(<g∇2⊥ - + 2∇4⊥ -) = N (-; ∇⊥ -; ∇2⊥ -) :
3;
(112)
Here N denotes all terms which are nonlinear in - and its derivatives. As noted in [158], the linear
terms in (112) are exactly the same as the linear terms appearing in the various model equations
discussed in Section 2.11 (the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation or the
Swift–Hohenberg equation at = 1). After a proper rescaling of space and time units, the pulled
front velocity v∗ for the Rayleigh–Taylor fronts in the thin 3lm experiments is therefore just given
by Eq. (88).
The experimentally observed front speed was indeed found to be consistent with the pulled speed
∗
v within the uncertainty of the experimental measurements and the theoretical estimate (which
was of order 30%). In addition, there seemed to be a clear trend for the front patterns to have
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
a wavelength some 5% smaller than the wavelength corresponding to the most unstable mode. This
is also what would be expected from the results for pulled fronts with the above linear dispersion
relation.
Although the results of the experiments thus seem to indicate that the Rayleigh–Taylor fronts in
the experiment are examples of pulled fronts, we do want to express an important caveat: just like
the bifurcation to hexagonal convection patterns is generally subcritical (except when a symmetry
imposes the equations to be invariant under a change of sign of the dynamical 3elds), the bifurcation
to hexagonal Rayleigh–Taylor patterns should be expected to have a subcritical character too; indeed
this is indicated by the weakly nonlinear results in [158]. This suggests that when the instability
is weak, one will actually get pushed rather than pulled fronts. However, to my knowledge this
possibility has neither been explored theoretically nor experimentally.
3.3. Pearling, pinching and the propagating Rayleigh instability
Another instability with which Rayleigh’s name is associated is the instability of a cylindrical
body of Kuid [78]. In daily life, we encounter this phenomenon when a stream of water from a tap
breaks up into drops or when the paint on a thin wire or hair of a brush breaks up into droplets.
The instability is caused by the surface tension: for a “peristaltic” perturbation of the radius of the
cylindrical Kuid, the capillary pressure due to surface tension is enhanced in the narrower region and
reduced in the wider regions; this pressure diDerence tends to enhance the perturbation even more.
In tubular membranes, the instability normally does not arise, since membranes generally have a
small surface tension but a high bending rigidity, which measures the resistance of the membrane
against changes in the curvature. However, Bar-Ziv and Moses [26] observed that when they applied
laser tweezers to their tubular membranes, a sinusoidal instability developed, see Fig. 25(a). This
instability propagated out at a constant velocity in both directions from the point of application of the
tweezers. Presumably, the instability is due to the fact that the tweezers pull some lipid molecules
into the trap, and that as a result the surface tension increases, rendering the cylindrical geometry
unstable. Indeed, as pointed out in [26], for a tube with surface tension = and bending modulus >,
the free energy F can be written as
F = dS[= + 2>H 2 ] :
(113)
Here H is the mean curvature and the integral is over the surface S. For a tube with cylindrical
symmetry and the z-coordinate along the axis, we can write the terms in F in terms of r(z) as
dS = d z 2r
1 + rz2 ;
H=
rzz
1
−
:
(1 + rz2 )3=2 r(1 + rz2 )1=2
(114)
As pointed out by Bar-Ziv and Moses a tube of constant radius R exhibits the Rayleigh instability
when the surface tension is above a critical value =c ≈ 3>=R2 . Above this value the free energy is
lower for a periodically modulated tube than for a tube of constant radius. As > → 0 one recovers
the pure Rayleigh instability. Thus, the experimental scenario is that the laser tweezers pull in the
lipid molecules, and that the increase in surface tension that this entails is large enough for the
membrane to become unstable.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
105
(b)
Fig. 25. (a) Photographs showing the pearling front along a tubular membrane formed by lipid bilayers in the experiments
of Bar-Ziv et al. [26,27], as it propagates out from the bright laser spot on the right. In this experiment the dimensionless
control parameter = (= − =c )==c = 6:5. From top to bottom the times are 0.14, 0.68, 0.86, 1.04 and 1:22 s after the laser
was turned on. The arrow indicates the leading edge as it was determined by the authors. The height of each photographs
corresponds to a distance of about 10 m. (b) Results by Powers and Goldstein [356] for the simpli3ed model (115) for
the pearling instability. The top panel shows four snapshots of the pearling obtained from numerical simulation of the
model. The higher snapshots are obtained at later times, so the front is traveling to the left. Bottom panel: dimensionless
front velocity as a function of the ratio of surface tension and bending modulus. The solid line denotes the pulled front
velocity v∗ , the dots the data obtained in the numerical simulations. Note that the data points lie slightly below the curve
for v∗ . Undoubtedly, this is due to the slow convergence of the front speed to the asymptotic speed v∗ .
The pearling instability through propagation of the Rayleigh fronts that the experiments suggest
have been studied theoretically by Powers and Goldstein and co-workers [356,185], who pointed
out that in the approximation that the Kow pro3le through the tube remains a parabolic Poiseuille
pro3le, 56 the dynamical equation for the tubular interface becomes
F
1
9 z r 4 9z
;
(115)
9t r 2 =
4;
r
where F is the interfacial free energy given in Eq. (113) above. The numerical results from [356]
for this equation are shown in Fig. 25(b). The top panel shows four snapshots as the Rayleigh
front propagates along the tube, generating strongly nonlinear bead-like undulations behind it. In the
lower panel the velocity of these fronts is compared to the pulled front velocity v∗ for this model
56
This approximation, which is in the spirit of the the thin-3lm equations described in the previous section, is justi3ed
in the limit in which distortion of the tube diameter happens on length scales much longer than the tube diameter. This is
not really true for the Rayleigh instability. Nevertheless, the equation is expected to capture the essentials of the pearling
instability.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 26. Experimentally measured values of the wavenumber of the pearling instability as a function of the dimensionless
control parameter = (= − =c )==c in the experiments of Bar-Ziv et al. [27]. The line labeled MSC is the wavenumber
selected by the pulled front in the model discussed in the text.
(full line) as a function of the dimensionless parameter R2 ==>. Note that this velocity vanishes at
a critical value—as already noted above, below this value the cylindrical tube is stable. Note that
the data points fall slightly below the pulled speed v∗ . Although not enough data are provided in
[356] to check this explicitly, this is undoubtedly due to the slow convergence of the front speed
to the asymptotic value v∗ —as mentioned before, numerical data at long but 3nite times will alway
approach v∗ very slowly from below, and indeed a discrepancy like the one in this plot is what one
often encounters if a careful extrapolation of the speed to its asymptotic value is not made.
It is instructive to realize that although the nonlinear behavior of this pearling model is quite
diDerent from those encountered before, the dispersion relation for small linear perturbations of the
radius is not! Note that when F is expanded to quadratic order in the deviations from a cylinder
shape, the highest derivative terms it contains are of order rzz2 . Upon taking the functional derivative
in (115), we immediately see that the dispersion relation !(k) is a polynomial in k of sixth degree
[185,27]. Thus, from the perspective of the linear dispersion relation and the calculation of v∗ , the
equation can be viewed as an interesting generalization to sixth order of the fourth order models
equations of Section 2.11!
In more recent experiments, Bar-Ziv et al. [27] have tested the predictions from the above theory
in detail, both for the propagation velocity and for the wavelength of the pattern selected by the
front. Their data for the pattern wavenumber as a function of the dimensionless control parameter
= (= − =c )==c are reproduced in Fig. 26. The full line in this plot (labeled MSC for “Marginal
Stability Criterion”) shows that the wavenumber of the pattern selected by a pulled front in this model
is slightly larger than the wavenumber corresponding to the fastest growing mode. The experimental
data are consistent with this trend although they lie somewhat above the predicted values. Whether
this slight discrepancy is real is not clear. In any case, however, the experiment illustrates an
important point: the data shown in Fig. 26 cover a large range of values of : if one knows the
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
107
(b)
Fig. 27. Results from the full hydrodynamical simulations of Powers et al. [357] of the propagating Rayleigh instability
in the absence of a bending modulus. (a) Sequence of drop shapes for viscosity ratio ≡ ;inner =;outer = 0:05 at times
tn = 6:67n;outer R==, n = 1; 2; 3; : : : ; 15 from top to bottom. The evolution of each connected component was computed
independently of the others. (b) The dimensionless Rayleigh front velocity as a function of the viscosity ratio . For
¿ 1 the smoothly propagating front behavior is lost.
dispersion relation for a given problem, predictions for the properties of a pulled front can be made
for any value of the parameters, not just close to threshold (1) where an amplitude equation
description with F-KPP-type fronts might be appropriate.
Although the driving force is diDerent—gravity in one case, surface tension in the other—there are
some clear similarities between these propagating Rayleigh fronts and the Rayleigh–Taylor fronts in
the thin 3lm equations discussed in the previous section. The analogy even extends to the following
issue: as we remarked at the end of our discussion of the Rayleigh–Taylor fronts, that instability is
expected to be (weakly?) subcritical; therefore when the linear instability is weak, one expects there
to be a transition to pushed fronts. The same might happen here: as pointed out by Bar-Ziv and
Moses [26], there is a small regime where the transition to modulated states of the tube is subcritical.
In this regime, one might therefore also expect pushed front solutions, but to my knowledge this
issue has not been studied here either.
Powers and co-workers [357] have also studied the pure Rayleigh instability (without bending
rigidity for the interface) numerically using the full hydrodynamic equations using a boundary integral
technique. As their calculations reproduced in Fig. 27(a) illustrate, in this case droplets pinch oD
when the viscosity ;inner of the Kuid inside the tube is smaller than ;outer of the outer Kuid.
Fig. 27(b) shows their data for the front velocity as a function of the viscosity contrast =
;inner =;outer . The full line is again the pulled front speed v∗ for the full problem, while the symbols
mark the numerical data. At the smallest values of these are some 30% lower than v∗ , but there
is every reason to believe that this is again due to 3nite simulation time and system size. Provided
this is true, these simulations are one of the most convincing ones that illustrate that pulled fronts
have no other choice than to propagate for large times with the linear spreading speed, even if the
dynamics behind them is highly nonlinear—what could be more nonlinear and nontrivial than the
pinching oD of a droplet?
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 28. (a) Snapshot of a convection pattern in a rotating Rayleigh–BIenard cell in the Kuppers–Lorz unstable regime, made
with the shadowgraph technique. One identi3es domains with essentially three orientations of rolls. From Bodenschatz
et al. [51]. (b) A snapshot of the spatial pattern in the statistically steady state in simulations of Tu and Cross [407] of
Eq. (116) with g+ = 2 and g− = 0:5. The dark, grey and light regions represent the domains occupied by A1 , A2 and A3 ,
respectively. The occupancy is determined by which one of the three amplitudes is largest.
3.4. Spontaneous front formation and chaotic domain structures in rotating Rayleigh–BIenard
convection
In Section 3.1 we already encountered a Rayleigh–BIenard experiment: a Kuid heated from below
exhibits a transition to convection patterns beyond some critical value of the temperature diDerence
between the top and bottom plate. When a Rayleigh–BIenard cell is rotated as well, the interplay
between the thermal buoyancy and the rotation-induced Coriolis force gives rise to a whole plethora
of new eDects, including traveling wave patterns [148,240,415]. One of the novel phenomena is the
so-called Kuppers–Lorz instability [242]: above a critical rotation rate /c , a standard stripe pattern
of straight rolls looses stability to a stripe patterns which make an angle of about 2=3 with the
original ones. However, the new pattern that emerges is in turn unstable to stripes which make
again an angle of about 2=3 with it, and so on: no homogeneous stationary stable pattern exists.
In practice, the systems settles into a statistical steady state of domains of stripes of roughly three
orientations, separated by domains walls or fronts invading these domains: new domains are created
and invaded incessantly. A snapshot from an experiment is shown in Fig. 28(a).
The interpretation of this statistical state was put forward by Tu and Cross [407]. They analyzed
the amplitude equations for the three roll amplitudes A1 , A2 and A3 in this system,
9t A1 = 92x1 A1 + A1 (1 − A21 − g− A22 − g+ A23 ) ;
9t A2 = 92x2 A2 + A2 (1 − A22 − g− A23 − g+ A21 ) ;
9t A3 = 92x3 A2 + A3 (1 − A23 − g− A21 − g+ A22 ) ;
(116)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
109
(b)
Fig. 29. (a) The front velocity v versus g− for various values of g+ as determined numerically by Tu and Cross [407]
∗
from the one-dimensional equations (120). The dotted line indicates the pulled front velocity vKL
given in (117). For
suRciently small g− the fronts are pulled. From [407]. (b) The inverse correlation length and inverse correlation time
T (de3ned in terms of the amplitude correlation functions) as a function of g− in the numerical simulations for g+ = 2.
Here 1, 2 and 3 label the three orientations whose normals ê1 , ê2 and ê3 make an angle of 2=3
with each other, and xi are the spatial coordinates projected along these directions, 57 xi = êi · r. The
amplitudes in Eq. (116) are taken to be real; this means that we can study the competition between
domains but not the variations of the wavelength of the convection patterns in each domain.
For g− = g+ this system of equations is symmetric in all the amplitudes Ai , and can be derived
from a Lyapunov function. This implies that the dynamics is then relaxational, the dynamics tends
to drive the system to the minimum of the Lyapunov function. 58 However, for g− = g+ (without
loss of generality one can take g− ¡ g+ ) the system has a cyclic permutation symmetry only and it
is non-potential: it cannot be derived from a Lyaponov functional. For g+ large and g− small, the
A2 amplitude is strongly suppressed by the A1 amplitude but not vice versa, so it is intuitively clear
that an A1 domain will invade an A2 domain, etc.
The snapshot of the simulation of the above equations [407] shown in Fig. 28(b) illustrates that
at any given time, the pattern consists of domains of three roll-orientations, separated by domain
walls or fronts: the dark domains invade the grey ones, the grey ones the light ones, and the light
ones the dark ones. Note also the similarity with the experimental picture of Fig. 28(a).
When Tu and Cross [407] analyzed the statistical properties of these steady states, a surprising
feature was found: while the Kuppers–Lorz instability sets in when g− decreases below 1, the
correlation length and correlation time T did not diverge as g− ↑ 1—see Fig. 29(b). The clue to
understanding this was identi3ed [407] to be the behavior of the fronts. For g− ¡ 1 a homogeneous
state is linearly unstable to the growth of one of the other two amplitudes; indeed, it is easy to see
57
The geometrical projection factors in 9xi = êi · 9r in the gradient terms in (116)–(117) are precisely those which are
responsible for the stripe-type appearance of fronts or domain walls at the edge of a hexagonal pattern—see footnote 55.
58
By saying that a system has a Lyaponov function F, we mean that it can be written in the form 9t A=−F=A, where
the term on the right-hand side is a functional derivative. An example is given in Eq. (120) below, and we also brieKy
encountered a Lyapunov function
in the discussion of the the Cahn–Hilliard equation in Section 2.11.3. The derivative
form implies that dF=dt = − (F=A)2 6 0 so that F either decreases under the dynamics or stays constant.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
from Eq. (116) that the linear spreading speed of an A1 perturbation into a domain where A2 = 1
equals
∗
vKL
= 2 1 − g− :
(117)
On the other hand, even though an A2 domain is linearly stable to small perturbations in all the
amplitudes, for 1 ¡ g− ¡ g+ it is only metastable: an A1 domain will invade an A2 domain even
though it is linearly stable. We can show this by using the curious feature that two-mode subdynamics
of these equations is actually of potential nature, even though the full three-mode dynamics is not.
Let us consider the one-dimensional dynamics of two modes only,
9t A1 = cos2 1 92x A1 + A1 (1 − A21 − g− A22 ) ;
(118)
9t A2 = cos2 2 92x A2 + A2 (1 − A22 − g+ A21 ) :
(119)
Here 1 and 2 are the angles between the x-axis and the vectors ê1 and ê2 . Upon transforming to
√
√
the variables AV 1 = g+ A1 and AV 2 = g− A2 we can write the equations in the potential form
F
9AV 1
=−
;
9t
AV 1
F
9AV 2
=−
;
9t
AV 2
(120)
where the “free energy”, which plays the role of a Lyapunov functional, is given by
1 V4
1 V4 V2 V2
F = 12 d x cos2 1 (9x AV 1 )2 − AV 21 +
A1 + cos2 2 (9x AV 2 )2 − AV 22 +
A2 + A1 A2 : (121)
2g+
2g−
√
√
For g1 ¿ 1, this free energy functional has local minima at AV 1 = g+ , AV 2 =0 and AV 1 =0, AV 2 = g− .
For g− ¡ 1 the second point becomes a saddle, in agreement with the fact that the state A2 state is
unstable to the growth of the A1 -mode. However, even though the free energy has two local minima
for g− ¿ 1, for any g− ¡ g+ the second one corresponding to the nonzero A2 -mode always has a
higher free energy than the 3rst one, which corresponds with the A1 -phase. These two minima are
separated by a free energy barrier (very much like two stable phases near a 3rst order transition), but
the relaxational nature of this subdynamics implies that any initial condition which corresponds with
an A1 -domain on the left and an A2 -domain on the right will develop into a coherent front which
moves to the right. Such “bistable” fronts which connect two linearly stable states are well known
for such types of equations; they are like pushed fronts and their speed approaches an asymptotic
value exponentially fast.
Indeed, as Tu and Cross found—see Fig. 29(a)— the scenario which emerges is that for any
g− ¡ g+ the dynamical attractor is a statistically steady chaotic domain state. If g− is reduced
below g+ , the invasion of domains is due to pushed fronts up to some critical value of g− below
which the fronts are pulled. 59 The Kuppers–Lorz instability is therefore not only a nice illustration
of how fronts propagating into an unstable state can be generated dynamically, but it is also one of
the few realistic examples we know of where the front propagation mechanism changes from pushed
59
∗
Note that the numerical data-points in the pulled regime in Fig. 29 are slightly below vKL
; as always, this is a sign
of the slow power law convergence to the asymptotic pulled front speed.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
111
to pulled upon changing a parameter. In this particular experiment this parameter can even be tuned
easily by changing the rotation rate. 60
We 3nally note that as as we will discuss in Section 5, pulled fronts are not amenable to the
usual sharp interface approximation or moving boundary approximation. Whether this gives rise
to any noticeable diDerence between the transient domains in the pushed regime just above the
Kuppers–Lorz instability and those in the pulled regime at higher rotation rates, has apparently not
been explored. However, I consider it unlikely that it does, since as Fig. 29(a) shows, there is only
a large separation of scales between the front thickness and the domain size in the regime near
threshold.
3.5. Propagating discharge fronts: streamers
When the electric 3eld is large enough, free electrons in a gas accumulate suRcient energy
inbetween collisions that they can knock out an electron from a neutral gas molecule in a collision. In
air, the ionization of nitrogen is dominant, and in this case the ionization reaction can be summarized
as e− + N2 ⇒ 2e− + N2+ . This type of avalanche phenomenon can naturally lead to to the formation
of discharge patterns (sparks!) whose dynamics is dominated by the propagation of fronts into the
unstable non-ionized state, as is illustrated in Fig. 30. Figures (a) and (b) show two snapshots of
simulations of Vitello et al. [433] of the formation of a discharge pattern in nitrogen between two
planar electrodes across which a potential diDerence of 25 kV is applied. Initially, at time t = 0, the
gas between the electrodes is non-ionized, except for a small region near the upper electrode. Due
to the large 3eld, the electrons immediately get accelerated downward into the non-ionized region,
ionizing neutral molecules along the way. Figs. 30(a) and (b) show the electron density level lines
4.75 and 5:5 ns later. The regions inside the 3nger-like regions of these so-called streamer patterns
are weakly ionized plasmas; the regions where the level lines crowd mark the zones where the
electron density drops quickly to a very small value, and where most of the ionization takes place.
The negative space–charge in this transition zone eDectively shields the outer 3eld in the non-ionized
region from that in the streamer body where both the space-charge and the 3eld are small.
The propagation of streamer patterns is well described by the following set of equations for the
electron density ne , the ion density n+ and the electric 3eld E [142,433],
9t ne = "0 |ne @e E|e−E0 =|E| − ∇ · (−ne @e E − De ∇ne ) ;
(122)
9t n+ = "0 |ne @e E|e−E0 =|E| ;
(123)
∇·E=
e
(n+ − ne ) :
0
(124)
The 3rst term on the right-hand side of the 3rst two equations is the impact ionization term. As @e
is the electron mobility, −@e E is the electron velocity, so the prefactor is just the impact rate of
electrons on the ions, while the exponential factor accounts for the ionization rate as a function of
60
For fronts between two locally stable states the front velocity will behave linearly in g+ − g− for g− ≈ g+ . Since the
domain dynamics is driven by the front motion, we therefore expect the correlation length and time scale T to scale as
=T ∼ (g+ − g− ) close to the point g− = g+ . The correlation length in Fig. 29(b) does seem to vanish indeed faster
than T , but there are insuRcient data to test this hypothesis.
112
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
(a)
(b)
(c)
Fig. 30. (a) and (b): results of the numerical simulations of Vitello et al. [433] of the full three-dimensional simulation of
Eqs. (122) – (124) for parameters corresponding to nitrogen gas and a potential diDerence between the planar electrodes
of 25 kV. Each line indicates an increase of ne by a factor of 10. Shown are snapshots of the electron density 4.75
and 5:5 ns after an initial ionization seed was placed at the upper electrode. (c) Space-time plot of a simulation of the
equations by Ebert et al. [142] for the case of a planar streamer front in the case D = De "0 =(@e E0 ) = 1 [142]. Upper
panel: the dimensionless electron density + = ne e=(0 "0 E0 ) as a function of time and space. Lower panel: the electric 3eld
measured in units of E0 . The initial condition was a homogeneous 3eld E = −E0 and a small, charge-neutral ionized
region with Gaussian electron and ion density around x = 55. The front propagating to the right in the same direction
∗
as the electron drift velocity is called a negative streamer front; it is a pulled front with asymptotic speed vstr
(=2.21 in
dimensionless units), given by Eq. (125). The front propagating slowly to the left is a pushed positive streamer front.
the 3eld—small 3elds give a small ionization rate, as the electron speed at impact is small. The other
two terms on the right-hand side of (122) account for the drift and diDusion of the electrons. Similar
terms are absent in the equation for the ion density n+ since on the time scales of interest drift
and diDusion of the much heavier ions can be ignored. Finally, Eq. (124) is the Maxwell equation
relating the 3eld strength to the charge density.
Clearly the impact ionization term in these equations makes the non-ionized state with ne = n+ = 0
linearly unstable. Moreover, since a nonzero charge density tends to screen the 3eld, the 3eld is
largest just ahead of a front. The exponential factor in the impact ionization term is an increasing
function of the 3eld strength, hence the growth is gradually decreased from a large value ahead of
the front to a small value in the plasma behind the front where the 3eld is screened. At 3rst sight, it
may therefore come as no surprise that discharge fronts which propagate in the same direction as
the electron drift velocity—so-called negative streamers—are examples of pulled fronts [141,142].
Now, apart from the drift term, Eq. (122) has, to linear order in ne , the same terms as the F-KPP
equation (1), a linear growth term and a diDusion term. We therefore immediately obtain the asymptotic speed of planar streamer fronts
as
∗
= vstr
= −@e E + + 2 De |@e E + |e−E0 =E + ;
(125)
vstr
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
113
where E + is the value of the 3eld just ahead of the front. This prediction has been fully con3rmed
by extensive numerical simulations by Ebert et al. [141,142], an example of which is shown in
Fig. 30(c).
The streamer problem is very instructive from a more general point of view. First of all, in the
F-KPP equation, it is rigorously known [16,35,37] that one is always in the pulled regime if the
growth function f()= ¡ f |=0 . The streamer problem nicely illustrates that such simple results
generally do not hold in more complicated cases. As we mentioned above, the ionization term in
the streamer equations rapidly decreases when the 3eld strength drops; hence the ionization rate
per electron "0 @e |E|exp − E0 =|E|, which is the analog of the growth ratio f()= in the F-KPP
equation, decreases monotonically from the front side to the back side of the front. In spite of this,
for each set of parameter values, the streamer equations not only admit pulled negative streamer
front solutions, but also pushed positive streamer front solutions which propagate in the opposite
direction. Fig. 30(c) illustrates this. The reason for this is that streamer front propagation arises
through the interplay of ionization and screening of the 3eld. Screening is a nonlocal phenomenon
mediated through Eq. (124), and so the monotonic behavior of the 3eld dependence of the local
ionization term is just half of the story.
Secondly, the negative streamer patterns like those of Figs. 30(a,b) are nice examples of interfacelike growth patterns whose dynamics is associated with propagating pulled fronts. Indeed, if we write
the electric 3eld in terms of the electrical potential , as E = −∇,, we see that outside the streamer
body, where the charge density is negligible, the potential obeys to a good approximation the Laplace
equation ∇2 , = 0. Moreover, Eq. (125) shows that to a good approximation the normal velocity of
a streamer equals @e ∇,. These are precisely the two equations for viscous 3ngering [44,250,338]
or for thermal plumes in two dimensions [451], so if we think of the streamer pattern as a moving
interface, we expect their dynamics to have a number of similarities with the dynamics of viscous
3ngers or plumes [141,428].
However, the story is not that simple! The problem is that pulled fronts cannot straightforwardly
be mapped onto a moving boundary problem, even when their width is much smaller than the
pattern scale [as is clearly the case for the streamer patterns of Figs. 30(a,b)]. As we will discuss
more generally in Section 5, the fact that the dynamically important region of pulled fronts is ahead
of the front itself entails not only a power law convergence to the asymptotic speed but also a
breakdown of the standard moving boundary approximation. The precise implications of this for
streamers are still under active investigation—very much like dendrites, they do show a tip-splitting
instability [18], but the dispersion relation of small perturbations of a planar discharge front does
appear to be diDerent from what one would expect based on the analogy with viscous 3ngering
or dendrites.
3.6. Propagating step fronts during debunching of surface steps
When a crystal is cut with an angle slightly diDerent from one of the principle crystal facets, the
resulting “vicinal surface” contains a lot of steps. The predominant mechanism during vapor growth
is then that adatoms which have landed on the surface diDuse towards these steps and attach there.
The crystal growth is thus accompanied by the propagation of steps along the surface.
It has been known since long that growing mono-atomic steps can “bunch”: instead of staying
equidistant, on average, they bunch together and form macro-steps. Many of these and other step
114
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 31. ReKection electron microscopy image of a vicinal Si(111) surface 10 s after the a change of direction of the
current direction; a series of steps “peel oD” from a large macro-step in a characteristic pattern: steps “debunch” from
the macro-step that is visible as the thick line meandering through the middle of the 3gure. From Latyshev et al. [253].
(b) Snapshot of a Monte Carlo simulation by Kandel and Weeks [215] of the stochastic version of the step model (127).
Note the similarity with the experimental observations shown in panel 3gure (a).
instabilities can be understood in terms of the following simple model of step Kow [42,168,214,411],
9t Xn = f+ (Wn ) + f− (Wn−1 ) + 292y Xn :
(126)
Here Xn (y; t) is the position of the nth step measured along y, the coordinate along the step.
The terms f+ (Wn ) and f− (Wn−1 ) describe the growth due to attachment of atoms from the terrace
of width Wn = Xn+1 − Xn in front of the step and the terrace of width Wn−1 just behind the step.
The last term is a curvature correction analogous to the surface tension type term that occurs in
almost all interfacial problems. Note that from a mathematical point of view, Eq. (126) is of mixed
character: it is a partial diDerential equation with respect to the y-variable, but a diDerence equation
with respect to the step index n.
A straightforward stability analysis shows that an equidistant array of steps (all Wn =W ) is unstable
when the attachment kinetics is such that steps predominantly incorporate atoms from the terrace
(W ) ¿ f (W ). This is in full agreement with the intuitive idea that when a step
behind it, i.e. if f−
+
lags a bit behind, the terrace behind it becomes smaller and hence the step will capture fewer atoms
from this terrace. 61 This instability leads to step bunching in the nonlinear regime.
It has been found in various experiments that application of a current along the surface can induce
very complex patterns. Indeed, one of the new phenomena that occurs is shown in Fig. 31(a): right
after reversal of the current, steps “peel oD” from a macro-step in a characteristic pattern.
This experimentally observed “debunching” behavior is not found in the above model for the
step dynamics, but Kandel and Weeks [215] have shown that it can be understood in terms of the
following extension of it. The shortcoming of model (126) is that the growth functions f± (W ) only
depend on the terrace width right ahead of and behind the step. However, when steps bunch they
61
Early work on step bunching by Frank [168] described it on a coarse-grained scale in terms of the Burgers equation.
If the step velocity decreases as a function of step density, the well-known formation of shocks in the Burgers equation
implies step bunching. From this perspective, the phenomenon is similar to the formation of traRc jams in elementary
models for traRc Kow.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
115
get so close that this simple approximation breaks down: capture of adatoms by steps in a bunch is
strongly suppressed because steps cannot move any closer. Instead, surface diDusion over the entire
bunch becomes more probable. Kandel and Weeks [215] therefore replace (126) by
9t Xn = f+ (Zn(f ) ) + f− (Zn(b) ) + 292y Xn ;
(127)
Zn(f )
(Zn(b) )
for Wn ¿ Wmin , while 9t Xn = 0 for Wn ¡ Wmin . Here
is the width of the 3rst terrace in
front of (behind of) the nth step that is larger than Wmin . Note that this introduces a dynamically
generated non-locality in the model: the terraces from which a step captures atoms depend on the
dynamical state of the step con3guration itself. As a result, as discussed in detail in [215], these
equations also lead to a debunching instability near the edge of a step bunch: one or a series of
steps can “peel oD” in a characteristic fashion from the bunch, while the other steps in the bunch
remain virtually immobilized together. A snapshot of a Monte Carlo simulation [215] of a stochastic
version of this model with
f− (W ) = k− W;
f+ (W ) = k+ W ;
(128)
is shown in Fig. 31(b). Clearly, the type of step patterns found in this model is remarkably similar
to that seen experimentally.
From our perspective, the interesting aspect is the arrays of steps—the almost parallel lines in the
3gure—which cross from one step bunch to the other. As we mentioned above, the step con3guration
near the edge of a bunch is unstable to the peeling oD of an array of steps, and so the propagation
of the detachment point can be viewed an an example of a front propagation into an unstable state
problem. Indeed, by linearizing the dynamical Eqs. (127) and (128) about an array of straight steps
separated by a distance ≈ Wmin and assuming a behavior for perturbations Xn of the form
Xn (y; t) ∼ e−(y−nE)++t
(129)
one obtains a linear dispersion equation of the form
+() = k+ (eE − 1) + k− (1 − e−E ) + 22 :
(130)
Note that the various terms in this expression do reKect the mixed character of the equation: the
exponential terms are characteristic of a 3nite diDerence equation, the last term of a partial diDer∗
ential equation. The linear spreading speed vdebunch
can of course be obtained straightforwardly by
∗
determining the minimum of the curve +()=. The comparison of vdebunch
with the simulation data
for various E is shown in Fig. 32. The data clearly show that the debunching point propagates along
the bunch with the linear spreading velocity, so apparently the debunching process is an example
of a pulled fronts. Pulled fronts were also found to describe certain aspects of the initial bunching
process in the original model (126) [214].
Recent experiments on this debunching instability [438,393] are consistent with the relation
between growth length and step spacing E found by Kandel and Weeks [215] but do not test
the predictions for the front velocity directly. More recent theoretical work [207] shows that a
strain-induced step bunching instability can be convective.
3.7. Spinodal decomposition in polymer mixtures
When a homogeneous mixture is quenched within the so-called spinodal region, the homogeneous
state is unstable to composition Kuctuations. The lowest free energy state towards which the system
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
∗
Fig. 32. Comparison of the prediction of vdebunch
(E) (full line) with the velocity
data (symbols) from the simulations
shown in Fig. 31(b). V0 is the initial step velocity V0 = (k+ + k− )W and Ec = 22=(k− − k+ ). From Kandel and Weeks
[215].
evolves at long times is one in which it is demixed into two homogeneous phases of diDerent
composition. On the way towards this demixed state the system is spatially inhomogeneous on
mesoscopic scales. This spatio-temporal demixing process is called spinodal decomposition [56,192].
At intermediate times the dynamics is normally dominated by motion of interfaces between domains in which the composition is close to one of the equilibrium compositions. The initial phase,
however, depend on the system under study. For most systems (like ordinary liquids), the Kuctuations are large enough that right after the quench these Kuctuations grow out due to the instability of
the bulk mode. Moreover, this regime is often diRcult to probe experimentally, since it happens on
too short time scales. For suRciently long polymers, however, the dynamics is slow enough that this
regime becomes experimentally accessible [440]. Moreover, the longer the polymers the smaller the
Kuctuations, so that the coarse-grained mean-3eld like models which have mostly been studied in the
literature become more appropriate for polymer systems. It is also conceivable that the short-time
dynamics is then sometimes dominated by the propagation of composition modulation fronts into
the unstable homogeneous state. If so, it is likely that such fronts typically start at the walls of the
sample, since a wall is normally preferentially wetted by one of the compositions [21].
A simple model equation for spinodal decomposition is the Cahn–Hilliard equation which we
already introduced in Section 2.11.3. The front propagation problem in this equation in the presence
of noise was studied from the above perspective by Liu and Goldenfeld [266] and by Ball and
Essery [21] who both found that the composition modulation fronts in this equation are pulled fronts
which lead to a incoherent pattern which continues to coarsen. 62 The simulations of Fig. 15 also
showed this.
Experimental evidence for the above surface-induced front-dominated spinodal decomposition
scenario in polymers was found by Jones et al. [208], but no direct quantitative comparison was
made with the predictions from the theory of front propagation.
62
The data points for the numerical front velocities obtained shown in [266] are as usual slightly below the value v∗ :
As we remarked already several times, this is true for almost all published data, and reKects the slow convergence of the
speed of a pulled front to its asymptotic value.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
117
We 3nally note that the competition between bulk growth and front propagation in a model with
a non-conserved order parameter was also studied in [289,409].
3.8. Dynamics of a superconducting front invading a normal state
The equilibrium and dynamical behavior or vortices in type II superconductors is a vast and
active 3eld with many rami3cations, which range from fundamental statistical physical studies to
applications [50]. Even in the area of vortex dynamics, interesting questions concerning the behavior
of propagating fronts separating domains with diDerent vortex properties (e.g. a diDerent density of
vortices) come up [137,285], but most of the issues that arise in this area are diDerent from those
which are the main focus of this paper. We therefore limit our discussion here to some interesting
theoretical 3ndings concerning fronts propagating into an unstable state of a type I superconductor,
which also raise some new fundamental questions. Some indirect evidence for such fronts have been
found from measurement of the magneto-optical response of thin 3lms [169] but the time-resolution
has been insuRcient to study the dynamics directly. Recent advances in magneto-optical techniques
[202,437] to visualize vortex patterns may open up the possibility to do so in the near future.
It is well known [394] that in so-called type I superconductors the normal state is linearly
unstable 63 to the superconducting state when the magnetic 3eld H is reduced below the critical
value Hc2 . For 3elds Hc2 ¡ H ¡ Hc , the normal state is linearly stable, but has a higher energy than
the superconducting state; the superconducting state can then only form through nucleation. The
barrier for this nucleation of superconducting domains vanishes as H approaches Hc2 from above.
The dynamical behavior that can result if a type I superconductor is quenched into this unstable state
was studied a number of years ago by Liu et al. [267] and by Frahm et al. [132,167], who pointed
out the analogy of these two regimes to the spinodal and nucleation regimes of phase separation
(see Section 3.7). Moreover, they drew attention to the fact that when the superconducting phase
propagates into the normal phase, the eDective long-wavelength equations show a strong analogy
with the diDusion equations describing diDusion limited growth. As a result, one expects these fronts
to have a similar Mullins–Sekerka like long-wavelength instability as crystal growth problems (see
[73,220,226,247,251,338,353] and Section 3.17).
In time-dependent Ginzburg–Landau theory the dynamics of a superconductor is governed by the
dynamical equation for the complex superconducting wave function (r; t) [132,167,132,394]
9
F
=− ∗ ;
9t
where the dimensionless free energy functional F is given by
˜ − A] |2 + (∇
˜ × A)2 } :
F = dr{−| |2 + 12 | |4 + |[(i>)−1 ∇
(131)
(132)
The Ginzburg–Landau parameter >=(T )=(T ) in this expression is the ratio of the √
penetration depth
and the coherence length ; type I superconductors are characterized by > ¡ 1= 2. Furthermore,
63
Tinkham [394] formulates this slightly diDerently, but if the the problem is formulated as a stability analysis, i.e., as
the problem of 3nding the growth rate e+t of the linear eigenmodes, one immediately sees that below Hc2 the normal state
is unstable.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 33. (a) Penetration of the propagation of a superconducting domain into a normal domain in numerical calculations
by Frahm et al. [167]. The penetration is visualized through the strength of the magnetic 3eld shown in greyscale, with
black corresponding to B ≈ Hc and white corresponding to B ≈ 0. The 3eld strength in the normal phase corresponds
to a value of about 0.57 Hc which corresponds to the nucleation regime where the normal phase is linearly stable, while
> = 0:3 and = = 0:1. Times are measured in units of the order parameter relaxation time [167,267]. (b) As in (a), but
now for a 3eld H = 0:28Hc which corresponds to the spinodal regime where the normal phase is locally unstable.
˜
A in (132) is the magnetic vector potential which is related to the magnetic 3eld B through B=∇×A;
using Amp^ere’s law and Ohm’s law, the dynamics of A is governed by
=9t A = [>−1 Im (
∗˜
˜ × (∇
˜ × A) :
∇ ) − | |2 A] − ∇
(133)
Here = is the dimensionless conductivity [167] and the term between square brackets in this expression is the supercurrent contribution in the Ginzburg–Landau formulation. The gradient term in this
term describes the generation of supercurrents in fronts or interfacial zones and near surfaces, while
the second term is the Meissner term which is responsible for Kux expulsion from the superconductor state where | |2 = 0 (note that it acts like a linear damping term for A which drives the 3eld
to zero).
Fig. 33 shows two examples of the simulations of Frahm et al. [167] of a superconducting state
˜ ×
propagating into the normal state of a two-dimensional sample with the magnetic 3eld B = ∇
A perpendicular to the sample. Since the magnetic 3eld is expelled from the superconductor a
good way to represent the dynamics is by plotting the strength of the magnetic 3eld in greyscale.
Fig. 33(a) shows an example of a superconducting front penetrating the normal state in the nucleation
regime (Hc2 ¡ H ¡ Hc ) where the normal state is linearly stable. Due to the Kux expulsion, the
superconducting region shows up as white in the 3gure, while the black zones illustrate the 3eld
enhancement due to the supercurrents in the interfacial region. Note also that the size of the protrusion
increases in time, in agreement with the analogy with diDusion-limited growth problems noted above
which suggests that the interface should be unstable on long length scales.
Fig. 33(b) shows that the dynamics is very diDerent when the front propagates into a linearly
unstable normal phase (the spinodal regime). In this regime the front itself appears to become very
complex, and the dynamics is accompanied by the occurrence of phase slips in the superconductor
order parameter (points in space and time where the amplitude | | vanishes so that the phase of
can change by a factor 2).
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
119
The scenario that one expects for superconductor fronts propagating into a normal state is somewhat similar to the one discussed for the quintic CGL equation in Section 2.11.6: for 3elds
Hc2 ¡ H ¡ Hc the behavior is subcritical, and the dynamics of the fronts separating the two phases
is like that of a bistable front or a pushed front. When upon reducing the 3eld the value Hc2 is
crossed, the normal state becomes linearly unstable—this corresponds to becoming positive in the
quintic CGL. The linear spreading speed v∗ then becomes nonzero, but clearly for 3elds just below
Hc2 the dynamically relevant fronts will remain pushed. The question then arises, however, whether
upon decreasing the 3eld even more the fronts may become pulled for some range of parameters,
just like it happens in the quintic CGL equation. If so, this would be of great advantage, since it
might provide a handle at calculating some of the properties of the complex patterns of Fig. 33(b)
analytically, just like the propagating Rayleigh instability of Section 3.3 allows one to obtain most
essential features of the pattern analytically.
To our knowledge, the question whether the full two-dimensional complex patterns in the spinodal
regime are governed in some cases by propagating pulled fronts has not been studied yet. Some
indication that a transition to pulled front propagation might be possible comes from the work of
Di Bartolo and Dorsey [123]. They studied the propagation of one-dimensional fronts in the absence
of the possibility of phase slip generation (as was taken to be real and equal to f below) and in
the case that the external 3eld in the normal phase vanishes. In dimensionless units, the equations
in this case reduce to
9t f = >−2 92x f − a2 f + f − f3 ;
(134)
= 9t a = 92x a − f2 a :
(135)
The authors studied the case in which the front propagates into the normal state with f = 0 and
a = a∞ . Since the magnetic 3eld H is this
∞ case is simply related to a by H = 9x a, and since a = 0
in the superconductor behind the front, −∞ H = a∞ is simply the amount of magnetic 3eld trapped
in the front region and the 3eld in the normal phase far ahead of the front vanishes. Di Bartolo and
Dorsey [123] found that fronts in this restricted equation are pulled for small enough values of a∞
but pushed for larger a∞ . However, because of the possibility of the generation of phase slips was
suppressed by taking f real, it is conceivable that fronts at some point are unstable to the splitting
oD of a localized normal region with some magnetic 3eld trapped and a faster growing growing
front with a smaller amount of trapped 3eld. Clearly, various issues regarding these superconductor
fronts still remain to be resolved.
We 3nally note that apart from the practical signi3cance of studying whether fronts might be pulled
in the two-dimensional case, this question is also relevant from the following perspective. As we will
discuss in Section 5, pulled fronts are not amenable to the usual moving boundary approximation.
The full implications of this are not known, so the superconductor front problem might be a good one
to explore this issue, if they admit pulled fronts. The only other realistic example known to me where
this issue appears to have immediate relevance for the pattern formation are the streamer discharge
patterns analyzed in Section 3.5. (See also the remark at the end of Section 3.4 on the Kuppers–Lorz
instability.) The recent advance in increasing the time resolution of magneto-optic techniques [437]
may open up the possibility that these dynamical issues will become experimentally relevant in the
near future.
nch
nce bra
turbule
planar Couette
pipe Poiseuille
thres
hold
bran
ch
"perturbbation amplitude"
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
"perturbbation amplitude"
120
lence
turbu
h
branc
planar Poiseuille
thr
es
ho
ld
bra
nc
h
5772
Reynolds number
Reynolds number
Fig. 34. Schematic illustration of the subcritical bifurcation structure of the transition to turbulence in Couette and Poiseuille
Kow [205]. For planar Couette Kow and pipe Poiseuille Kow (left plot), the laminar base Kow is linearly stable for any
3nite Reynolds number, but the threshold to turbulence decreases as Re−2 with some exponent 2 as Re → ∞ [80]. Planar
Poiseuille Kow is linearly unstable for Re ¿ 5772 but the transition is subcritical and the linear instability does not play
much of a role in practice for Re of order 2000 or less, where turbulence is often already observed in practice. Note
that the drawing is schematic only: although only a single curve is drawn in the left plot, there are important diDerences
between planar Couette Kow and pipe Poiseuille Kow.
3.9. Fronts separating laminar and turbulent regions in parallel shear =ows:
Couette and Poiseuille =ow
In this and the next three sections we will discuss hydrodynamic instabilities in which both front
dynamics and Kuctuation eDects or turbulence play an important role.
Two of the basic textbook examples of hydrodynamic Kow states are Couette Kow and Poiseuille
Kow. We have already encountered Couette Kow in Section 3.1 where we considered Kow between
two concentric cylinders. For that setup, Couette Kow refers to the basic laminar Kow state. The
many instabilities that are found in this system are due to the interplay of the inertial eDects and
the Coriolis force in a rotating system. In this section, we will be concerned with planar Couette
Kow, Kow between two plates sheared in opposite directions. Experimentally, this setup is realized
by moving a transparent plastic band between two glass plates; the Kuid in between is then sheared
by the plastic band [108,109,283]. In planar Couette Kow of a normal Newtonian Kuid, the basic
Kow state for Kow in the x-direction between plates separated by a distance 2d in the y direction
is simply a linear velocity pro3le vx = vplates y=d; thus the shear rate 9vx =9y is constant. The other
classic example of a shear Kow is Kow through a pipe or between plates (the planar case) driven
by a pressure gradient: Poiseuille Kow. In this case, the basic velocity pro3le is parabolic, so the
shear rate is zero on the center line and increases linearly towards the walls.
The control parameter for Couette Kow or Poiseuille Kow is the Reynolds number Re = Ud=4
where U is a typical velocity of the Kow (the velocity at the plates in the case of Couette Kow), d
the half spacing of the plates or the radius of the pipe, and 4 the kinematic viscosity of the Kuid.
For large enough Reynolds numbers, Couette Kow and Poiseuille Kow become turbulent. However,
the transition to turbulence in such systems is not a supercritical bifurcation, in which a nontrivial
mode grows gradually in amplitude beyond a threshold at which the basic Kow becomes unstable.
Instead, the transition is subcritical [80,190,205], as sketched in Fig. 34. This implies that over some
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
121
(b)
Fig. 35. Example of a turbulent spot in planar Couette Kow, taken from the review of Manneville and Dauchot [283]. The
seeding of the Kuid with small Kalliroscope Kakes makes the turbulent regions show up brightly, while laminar regions
remain dark (the Kakes are lined up by the shear gradient in laminar regions). The spreading of such turbulent spots
depends on the Reynolds number of the base Kow [108,109,283]. (b) Snapshot of a turbulent spot in the simulations of
Schumacher and Eckhardt [378]. The Kow is in the x-direction. On the right, a cross-section in the direction normal to
the planes is shown. See Fig. 36 for further details.
range of Re linearly stable laminar Kow can coexist with turbulent regions 64 (very much like near
a 3rst order phase transition). An example of a turbulent spot in a planar Couette experiment is
shown in Fig. 35(a). The competition between such turbulent and laminar regions as a function of
Reynolds number has been explored in detail only in the last decade [108,109,283].
Actually, as Fig. 34 indicates, although the subcritical behavior is common to all three cases, there
is an important conceptual diDerence as well: planar Poiseuille is linearly unstable beyond Re =5772
while planar Couette and pipe Poiseuille Kow are stable for any 3nite Reynolds number—the critical
amplitude in the latter cases decreases as Re−2 for Re → ∞ [80,190] and hence we can think of
this case as an instability which has been pushed to in3nity. In the latter two cases, the absence
of a true instability also implies that there cannot be any pulled fronts: the fronts that separate a
turbulent region from a laminar region must always be pushed.
For the case of planar Poiseuille Kow, one might wonder whether the propagation of a turbulent region into the laminar Kow state could correspond to a pulled front, as the situation is
somewhat similar to the one found for the quintic complex Ginzburg–Landau equation discussed in
Section 2.11.6. 65 However, a priori this possibility is already unlikely for realistic Reynolds numbers: since the generic behavior of planar Poiseuille Kow is so close to that of planar Couette and
pipe Poiseuille Kow, where fronts de3nitely have to be pushed, we similarly expect the fronts separating turbulent and laminar domains to be pushed as well. It was indeed found [116] that the
linear instability of the planar Poiseuille pro3le always is always convective, so that pulled fronts
could never propagate upstream. As we shall see below, however, the simpli3ed picture based on
a straightforward linear stability analysis of the unperturbed Kow does not suRce for studying the
spreading of a spot, because a coupling with induced cross Kow has to be taken into account.
64
It is interesting to note that also in turbulent superKuid pipe Kow, there are strong experimental indications for the
coexistence of turbulent and laminar domains [284].
65
In the quintic complex Ginzburg–Landau equation, expanding chaotic spots were observed in numerical simulations
in some parameter ranges [424]. Their behavior is very much like what one would expect intuitively for turbulent spots.
See footnote 18.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 36. Snapshots of the stream-wise velocity vx (z; y) as it spreads downward in the simulations of Schumacher and
Eckhardt [378] for Re = 200. Dotted lines denote contour lines for positive velocity values, solid lines the contour lines
with negative velocity values. Note that the turbulent region propagates downward with a roughly constant speed.
The point at issue here has recently been studied numerically for a closely related situation by
Schumacher and Eckhardt [378]. These authors performed direct simulations of the Navier–Stokes
equations for the case of shear Kow between planes. However, unlike in the case of plane Couette
Kow, for numerical convenience they imposed free-slip boundary conditions on the bounding plates
and drove the bulk Kow by bulk force which varies as a cosine in the y-direction normal to the
planes, and which forces the Kow in opposite directions near the two plates.
Fig. 35(b) shows a snapshot of a turbulent spot in the simulations, which evolved from a localized
perturbation of the laminar pro3le in the center of the simulation cell. Note that qualitative similarity
with the turbulent spots observed experimentally: Both spots shows a number of streaks in the
stream-wise (horizontal) direction. These streaks can also be recognized in cross-sections in the
y-direction perpendicular to the planes shown on the right.
The spreading of the turbulent spot in time in the “span-wise” (vertical) direction in these simulations is shown in Fig. 36. It is immediately obvious that the turbulent region spreads vertically
through some kind of front structure which advances with a rather well-de3ned speed. Moreover,
the local turbulent energy v2 averaged over the normal direction was found to decay exponentially
into the laminar region, very much as one would expect for a well-de3ned front solution and as has
been found in some incoherent regimes of the quintic CGL equation [424]. Since Schumacher and
Eckhardt found that the turbulent spot also gives rise to an overall “span-wise” outKow Uz in the
vertical direction, they analyzed the linear spreading velocity v∗ by linearizing the Kow equations
about a laminar state which is the sum of the base Kow and this outKow Uz [378], using the so-called
Orr–Sommerfeld equation. 66 The measured front speeds in the numerical simulations were about
66
The Orr–Sommerfeld equation is the linearized equation governing the stability of nearly parallel Kows. It is based
on linearizing the Navier–Stokes equations in the deviations about a base Kow pro3le U . [135]. In the inviscid limit the
equation reduces to the so-called Rayleigh stability equation used in some of the work discussed in Section 3.10 on the
formation of BIenard–Von Karman vortices.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
123
a factor 10 larger than the value of v∗ obtained this way, leaving no doubt that in the parameter
regime studied the span-wise spreading of a turbulent spot is not governed by a pulled front: These
fronts are pushed. An earlier analysis along the same lines of the spreading of turbulent spots in
plane Poiseuille Kow lead to the same conclusions [197].
We would like to stress that the above observations are just empirical: the simulations do give
strong indications that it does make sense to associate the expansion of a turbulent spot with a rather
well-de3ned coherent front, but, as we already pointed out in Section 2.7.3, not much is known about
the properties of such fronts. In fact, also in the quintic CGL equation, spreading of chaotic spots
was found to proceed with a more or less constant speed larger than v∗ (see Fig. 18(a)), but to our
knowledge it is a complete mystery why and how this happens, and whether there are still common
mechanisms at play.
From this perspective, it is very intriguing that the simulations show that the turbulence front has
a spatial decay rate which is about twice as large 67 as ∗ : these pushed turbulent fronts apparently
also fall oD with an exponential decay rate larger than ∗ , just as uniformly translating pushed fronts
or coherent pattern forming fronts do!
3.10. The convective instability in the wake of blu> bodies: the BIenard–Von Karman
vortex street
Another classic hydrodynamic instability which was analyzed by some of the founding fathers of
Kuid dynamics, BIenard and Von Karman, is the formation of a “vortex street” behind a cylinder
in a Kow: for Reynolds numbers Re less than about 4 the Kuid Kow around a cylinder is laminar.
When Re increases past this value, two symmetric eddies form behind the cylinder, but when Re
become larger than about 50, vortices are shed from the cylinder in an asymmetric pattern. This well
known instability is also of immediate technological interest, since the formation of these vortices
can cause serious damage to suspension bridges if they are in resonance with the eigenfrequencies
of the bridge. Much of the original work focused on the formation and dynamics of these vortices,
their spacing, etc., in other words on the well-developed strongly nonlinear Kow regime. In the last
two decades, it has become clear from experimental [286] and theoretical work [401,446], however,
that the onset of the instability is associated with a linear instability in the wake of the cylinder.
In 1984, Mathis et al. [286] studied the velocity Kuctuations in the region behind the cylinder
(the “wake” of the cylinder) using laser Doppler velocimetry. It was found that beyond a threshold
which for√long cylinders approaches a value Rec = 47, the root mean square velocity Kuctuations
grow as Re − Rec . This is already a strong indication that the transition to vortex formation is
associated with a supercritical (“forward”) bifurcation in the wake, and that the behavior there could
be modeled by a CGL type of amplitude equation.
Further evidence for this scenario came soon from an analysis of the stability of the Kow in
the wake [203,300–302,401,446]. In this region, the Kow is nearly parallel. It varies rapidly in the
y-direction normal to the cylinder and the overall Kow direction, but very slowly in the stream-wise
direction itself. If one then ignores in a 3rst approximation the variation of the Kow in the stream-wise
67
According to Fig. 8 of [378], in dimensionless units the exponential
decay rate ≈ 1:05 at Re = 300. From the
numerical values listed after Eq. (19) in this paper, one 3nds ∗ = =D ≈ 0:55 at Re = 300.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
x-direction, 68 one can then decompose the Kow perturbation in this direction in terms of Fourier
modes by writing
Kow 3eld ∼ f(y; k)eikx−i!(k)t ;
(136)
where the eigenfunction f(y; k) and the dispersion relation !(k) are determined by the Orr–
Sommerfeld equation. The latter is based on linearizing the Navier–Stokes equations in vy and
vx (x; y) − U (y) about a solution for the velocity pro3le vx = U (z) in the wake [135]. Once !(k) is
determined from the eigenvalue problem of the Orr–Sommerfeld equation (or the Rayleigh equation
to which it reduces in the inviscid limit), the linear spreading speed v∗ can easily be determined numerically. An analysis along these lines shows that the Kow becomes linearly convectively unstable
for value of Re around 20, and that for Re around 40 the wake becomes linearly absolutely unstable.
This means that for Re above this value, a pulled front in the wake will propagate upstream, towards
the cylinder. The frequency of the mode at the transition point, Re !(k ∗ ), then gives the frequency
of vortex shedding.
The above results imply the following scenario for the BIenard–Von Karman vortex instability. At
some critical Reynolds number Rec , which the experiments indicates to be about 47 [286], the region
behind the cylinder changes over√from convectively unstable to absolutely unstable. The saturation
of the velocity Kuctuations as Re − Rec indicates that one can describe the behavior close to
threshold with a cubic CGL equation for the variation of the unstable mode along the stream-wise
x-direction. According to the linear dynamics of the Kow equations in the wake, the instability
is convective below Rec and absolute above Rec : in the latter regime a perturbation also spreads
upstream towards the cylinder. If a perturbation of the wake Kow 3eld grows large enough that
nonlinearities become important, then the perturbation develops into a front. This front is pulled (as
fronts which propagate into the zero-amplitude state of the cubic CGL equation are always pulled)
and hence its velocity v∗ behaves just as discussed already above: for Re ¿ Rec the pulled front
moves upstream towards the cylinder. In this picture, the vortices arise as the highly nontrivial and
strongly nonlinear structures behind the pulled front! The situation is thus analogous to the one for
the propagating Rayleigh instability discussed in Section 3.3, where the propagating Rayleigh front
left behind pinching droplets.
For a detailed study and discussion of this scenario of vortex shedding, we refer to the recent work
by Pier and coworkers [344,345]. We also note that the ideas have been taken further by Leweke
and Provansal and coworkers: they show that secondary transitions in the wake of bluD bodies (like
a cylinder) in the regime 180 ¡ Re ¡ 350 are both qualitatively as well as quantitatively described
by a CGL equation for the behavior in the wake [5,261,262]. Some of the incoherent behavior of
the Kuctuations is attributed to a Benjamin–Feir instability [105,189] of the nonlinear mode.
68
At this point, of course, there is a strong connection with the discussion of so-called “global modes” discussed
in Section 6. In principle one should take the equation linearized about an overall Kow pattern U (y; x) whose spatial
x-dependence in the stream-wise direction is kept. Then the linearized equation has spatially dependent coeRcients, so
one cannot do a Fourier-type analysis. Usually, a WKB-type analysis of the resulting global mode is done instead at this
stage. In the limit where the region of instability is large enough, the latter type of analysis reduces to the local one used
here [84]. See Section 6 for further discussion of this point.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
125
3.11. Fronts and noise-sustained structures in convective systems I:
the Taylor–Couette system with through =ow
In almost all hydrodynamic systems with advection due to an overall Kow, front propagation is an
important ingredient of the dynamics. We already encountered an example in the previous section on
the BIenard–von Karman instability in the wake of a cylinder. There are also many pattern-forming
systems in which the primary bifurcation is a Hopf-bifurcation to traveling waves. The issue whether
the instability is convective or absolute then immediately arises, and if the instability is strong
enough that nonlinear saturation eDects play a role—as is generally the case near a supercritical
(forward) bifurcation—fronts often play an important role in the dynamics and in the emergence of
noise-sustained structures. E.g., the so-called “blinking” traveling wave state in binary mixtures can
be understood quantitatively in terms of fronts which propagate back and forth in the experimental
cell [104,162], and also incoherent pulse dynamics in traveling wave systems has been analyzed in
such terms [414]. In this and the next section, we focus on two examples where front dynamics is
intimately connected with the transition from coherent to incoherent pattern dynamics. In this section
we focus on an example of Kow-induced advection, and in the next section we discuss aspects of
the role of fronts in systems exhibiting a Hopf bifurcation to traveling wave states.
We already introduced the Taylor–Couette cell in Section 3.1: two concentric cylinders whose
“gap” between them is 3lled with a Kuid. When the inner cylinder is rotated, at some critical
rotation rate /c a stationary Taylor vortex pattern is formed; this pattern is periodic in the direction
along the axis of the cylinders. Babcock et al. [19,20] and Tsameret and Steinberg [404,405] studied
the behavior near threshold in the presence of a through Kow in the axial direction. 69 Such a
through Kow has two eDects. Firstly, it changes the onset of the instability to Taylor vortex patterns.
Secondly, and more importantly, it changes the nature of the instability, as the through Kow obviously
advects perturbations away from the inlet of the Kow at a 3nite rate. Hence just above the instability
threshold, when the growth is small, the system is only convectively unstable. This is illustrated
in the space–time diagram of Fig. 37(a), which show the evolution of a perturbation which was
initiated near the inlet. Clearly, the initial perturbation grows out and spreads (the pattern widens in
time) while being advected away down the axis. The fact that also the left Kank of the perturbation
is moving to the right—hence is retracting—con3rms that the system is convectively unstable in
this case.
By studying whether the center of such a wave packet grows or decays in the co-moving frame,
both groups have extracted the threshold c = [/c (Re) − /c (0)]=/c (0) as a function of the Reynolds
number Re of the through Kow. The results from Tsameret and Steinberg [405] are reproduced in
Fig. 37(b). The experimental variation of the threshold is in excellent agreement with the results of
a direct stability analysis of the Couette Kow with axial through Kow. For values of above the line
in the diagram, the Kow in an in3nitely long system is unstable to the formation of Taylor vortices.
If there were no noise in the system, one would not observe patterns in a 3nite system in the
convectively unstable regime just above the line in Fig. 38(b). In practice, there is of course always
some noise, and the type of patterns this gives rise to as is increased above c (Re) is illustrated
in Fig. 38(a) from Babcock et al. [19].
69
Similar eDects will happen in a Rayleigh–BIenard cell with through Kow [305], but this system does not appear to
have been studied experimentally in as much detail.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 37. The growth of a perturbation in the convectively unstable region of the phase diagram for Re = 6 and = 0:0477
( − c (Re) ≈ 0:032) in the experiments by Tsameret and Steinberg [405] on Taylor-Couette Kow with though Kow. The
pulse was generated by a sudden movement of the inlet boundary back and forth. Times are measured in units of the
vertical diDusion time $v (=d2 =4 with 4 the kinematic viscosity of the Kuid) and the distance along the axis is measured
in units of the gap spacing d. (b) The threshold to the convective instability c (Re) as a function of Reynolds number Re
in similar experiments by Tsameret and Steinberg [405]. The solid line marks the threshold according to a full theoretical
stability calculation, and the solid circles the threshold values determined experimentally by tracing the evolution of a
pulse generated near the inlet. The open squares are data obtained by extrapolation from a 3t to the CGL equation down
from higher values. At small enough values of Re, these agree to within experimental error.
(a)
(b)
Fig. 38. (a) Space–time plots of the patterns at Re = 3 for increasing values of in the experiments by Babcock et al.
[19], from bottom to top = 0:0347, 0.0632, 0.0822 and 0.1020. Only the 3rst quarter of the apparatus behind the inlet is
shown. (b) A longer space–time plot in the convectively unstable regime from the experiments of Tsameret and Steinberg
[405] at = 0:04 and Re = 3. The full vertical line marks the position of the front de3ned by the position where the
pattern amplitude reaches a 3xed value.
The lower panel corresponds to the case just above threshold ( − c ≈ 0:031). In this case,
incoherent Taylor vortex patterns are found a distance of order 25 down the cylinders away from
the inlet—small perturbations and Kuctuations near the inlet are ampli3ed while they are advected
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
127
(a)
(b)
(c)
Fig. 39. Various data from the experiments of Babcock et al. [19]. (a) (Upper left panel) Normalized mean squared width
+2 of the peak in the frequency spectrum as a function of at Re = 3 obtained from the measurement of the pattern
amplitude as a function of time at a position 100d behind the inlet (note that this is three times further from the inlet than
the region shown in Fig. 38(a)). The sharp transition at = 0:065 indicates the change from the small- regime where the
patterns are incoherent due to phase noise to a coherent large- regime where pattern are coherent. This value is therefore
associated with the transition from the convectively unstable to the absolutely unstable regime. (b) (Lower left panel)
Corresponding results for the CGL equation (137) with three diDerent noise levels, 10−6 (×), 10−5 (), and 10−4 ( ). The
CGL
dashed line locates the transition ca
= 0:071 [19]. (c) Stability diagram for axisymmetric Taylor vortex patterns. The
lower line and data points mark the line c (Re) where the instability sets in. The upper data points locate the transition
ca from the convectively unstable regime to the absolutely unstable regime as determined from the change-over from
CGL
coherent to incoherent patterns via measurements of the spectrum like those in (a), while the dashed line indicates ca
.
Between the lower line and the upper line, the system is convectively unstable, and noise-sustained structures like those
in the one but lowest panel of Fig. 38 occur. Open symbols indicate the boundaries where noise sustained structures occur
in simulations, from top to bottom, at z = 25d, 50d, and 100d. The dotted lines indicate estimates of these boundaries
from the deterministic CGL equation (137) with a 3xed value of A(z = 0) imposed.
◦
away, and only at this point have they grown large enough in size that they are measurable. Behind
this region, they saturate (the plot actually shows only the 3rst quarter of the total cell). In this
region, the combination of the Kuctuations and perturbations near the inlet and the convective eDects
give rise to an incoherent front-like state. As Fig. 38(b) from the other group [405] illustrates, the
eDective front position (de3ned by tracing the point where the pattern amplitude reaches a certain
level) is indeed slowly wandering back and forth on a longer time scale.
As we go up in Fig. 38(a), is increased. The pattern 3lls more and more of the cell, and at the
same time become more coherent, while the width of the front separating the two regions decreases.
Clearly, in the upper panel for = 0:102 the Taylor vortex pattern 3lls the whole cell, and one is
in the absolutely unstable regime. The pattern selection in this regime and near the transition was
studied numerically in [68,69,272,390].
The transition from the convectively unstable to the absolutely unstable regime is reKected in the
coherence of the pattern. A good way to quantify this is to measure the frequency of the pattern
at a 3xed position. The more coherent the pattern, the sharper the peak associated with the pattern
frequency (the position of the peak is simply 3xed by the wavelength and phase velocity of the
pattern). The upper panel of Fig. 39(a) shows the mean squared width +2 of the measured peak.
The experimental data on this system have been compared in detail [19,405] with the results of
numerical studies [318] and with the analytical predictions based on the amplitude equation. Just
above the threshold of supercritical (forward) bifurcation to a 3nite-wavelength pattern, the pattern
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dynamics can quite generally be described by an amplitude equation [105,189,193,435]. Because of
the advection of the patterns by the through Kow, the appropriate amplitude equation in this case is
the cubic CGL equation
$0 9t A + s0 9z A = 20 (1 + ic1 )92z A + (1 + ic0 )A − g0 (1 − ic3 )|A|2 A ;
(137)
where the second term on the left-hand side is the group velocity term associated with the advection
of the pattern, and where z is the 3xed coordinate frame along the axis of the cylinders. For an
impression of the various types of behavior of the incoherent structures in the CGL equation with
an advection term in a 3nite system with noise we refer to the work by Deissler [115–117,119],
Deissler and Kaneko 70 [118] and Proctor et al. [358]. In the present case, all parameters associated
with the terms linear in A in this equation were known directly from the stability analysis [318],
while c3 was determined via the standard method of calculating the nonlinear terms in an amplitude
expansion [318] (the parameter g0 only sets the amplitude scale and hence is not of importance in
comparing theory and experiment). This therefore allowed a direct comparison between theory and
experiment to be made with only one adjustable parameter, the eDect of the noise at the inlet. 71
Fig. 39(b) shows the results of simulations of Babcock et al. [19] in which only a random noise
term of strength E was added to Eq. (137). As can be seen, the behavior of the mean squared
width of the spectrum of A in the simulations, sampled at the same distance z = 100d as in the
experiments, quantitatively reproduces the experimental data of Fig. 39(a)! Moreover, it is seen that
the behavior is quite independent of the value of the noise strength E, so the comparison between
theory and experiment is eDectively made without adjusting any crucial parameter!
For the CGL equation (137), the transition from the nonlinearly convectively unstable to the
nonlinearly absolutely unstable regime can easily be calculated explicitly since the fronts are pulled.
For zero group velocity, s0 = 0, the linear spreading velocity v∗ is simply
0
∗
vCGL = 2
(1 + c12 ) ;
(138)
$0
compare Eq. (100) for the case 0 = 1, $0 = 1. In the presence of the advection term with s0 , a pulled
front connecting the A = 0 state on the left to the saturated state on the right moves with velocity
∗ . When this velocity is positive, the front is convected away and the system is convectively
s0 − vCGL
∗
unstable, while when s0 −vCGL
¡ 0 the front moves upstream to the inlet and the system is absolutely
unstable. The transition from the convectively unstable to the absolutely unstable regime therefore
∗
occurs when s0 − vCGL
= 0, i.e., for
ca =
s02 $20
:
420 (1 + c12 )
(139)
For the parameters corresponding to the experiments of Figs. 39(a) this gives the value marked with
the dashed line in panel (b)—the remarkable agreement con3rms that the transition from coherent,
virtually noiseless patterns at larger values of to noise-induced Kuctuating structures coincides with
the transition from the absolutely unstable to the convectively unstable regime.
70
Deissler and Kaneko [118,213] have introduced in particular the notion of velocity-dependent Lyapunov exponent.
This concept will be touched upon brieKy in Section 3.22.
71
Several features of the scaling behavior are actually captured by the deterministic equation if one takes the value of
A at the inlet at z = 0 3xed at a given value [19,405].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
129
Finally, Fig. 39(c) from [19] shows the full phase diagram of the present system as a function
of Re. The full symbols along the upper line mark the transition from the coherent patterns to the
incoherent noisy patterns determined experimentally from the spectrum, as in panel (a) for Re = 3,
and the dashed line marks ca according to (139). The open symbols denote the values of where
noise-sustained structures arise for three diDerent downstream values of z.
A detailed theoretical study of the Taylor–Couette system with through Kow, which goes beyond
the amplitude equation, can be found in [69,349,361]. Also the Rayleigh–BIenard system with through
Kow has been analyzed along similar lines [70,210].
In conclusion, these experiments on Taylor–Couette patterns in the presence of through Kow nicely
illustrate several conceptual issues: (i) the distinction between nonlinearly absolutely unstable and
convectively unstable regimes; (ii) the change from coherent to incoherent patterns that this transition
implies; (iii) the fact that in realistic systems true front structures arise where saturation behind the
front is important; (iv) since fronts in the cubic Complex Ginzburg–Landau equation are pulled,
all essential properties can still be calculated explicitly from the linear dispersion relation; (v) the
importance of noise sustained structures in general for convective systems.
The eDect of noise on convective systems has recently been studied systematically for the CGL
equation by Proctor et al. [358], who map out the full phase diagram (see also [272,390]). In line
with our discussion of the behavior of fronts in the CGL equation in Sections 2.11.5 and 2.11.6,
the behavior as a function of the control parameter depends strongly on whether or not the state
selected by the front is unstable to the Benjamin–Feir instability.
We 3nally note that recently the eDect of through Kow on chemical reactions has also been studied
experimentally and theoretically [211,290].
3.12. Fronts and noise-sustained structures in convective systems II: coherent and incoherent
sources and the heated wire experiment
In the previous section, we discussed the relation between noise-sustained structures in pattern
forming system in which an advection of the patterns is induced by externally imposing a Kow.
In pattern forming systems which exhibit a Hopf-bifurcation to spatially and temporally periodic
patterns, the ensuing traveling wave patterns intrinsically have a nonzero group velocity, and close
to the threshold the instability to a single mode is always convective. If the transition is supercritical (forward), then close to the threshold in the convectively unstable regime the emergence of
noise-sustained patterns is again intimately connected with the dynamics of pulled fronts.
We focus here on the discussion of defects in one-dimensional systems which exhibit a supercritical transition to traveling-wave patterns, as these are most intimately connected with the front
propagation issue of interest in this paper. One should note, however, that the motion of defects has
experimentally also been studied in great detail in binary mixtures [217,235]. The fact that transition
to traveling wave states in this system is subcritical rather than supercritical gives rise to a number
of interesting additional eDects, like the “locking” of a defect to the underlying period of the pattern
[46,352]. We refer to the papers by Kaplan and Steinberg [217] and Kolodner [235] for examples
and for an entry into the literature on these issues.
A new feature, in comparison with the discussion of the previous section, is that if the underlying system is spatially reKection symmetric, two types of traveling-wave states will be possible,
left-moving waves and right-moving waves. If each mode suppresses the other, the long-time
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dynamical state of the system is often dominated by sources and sinks. A source is a solution
which sends out left-moving waves to the left, and right-moving waves to the right, while a sink
absorbs a right-moving wave from the left and a left-moving wave from the right. Since sources
are the active generators of the traveling waves, their behavior is most important for the dynamics
of a traveling wave pattern. As we shall see, sources induce a sharp crossover from coherent to
incoherent dynamics, which is closely related to the one discussed in Section 3.11 above.
The appropriate amplitude equations for the pattern dynamics just above onset of a Hopf bifurcation to traveling wave patterns are [105]
$0 (9t AR + s0 9x AR ) = AR + 20 (1 + ic1 )92x AR − g0 (1 − ic3 )|AR |2 AR
− g2 (1 − ic2 )|AL |2 AR ;
(140)
$0 (9t AL − s0 9x AL ) = AL + 20 (1 + ic1 )92x AL − g0 (1 − ic3 )|AL |2 AL
− g2 (1 − ic2 )|AR |2 AL :
(141)
Here AR and AL are the amplitude of the right and left moving waves, and s0 is the linear group
velocity. 72 The terms on the 3rst line of each equation are the same for each individual mode as in
the single CGL equation (137) for the advected Taylor vortices. The last term on the second line
describes the coupling of the modes. We will be interested in the regime g2 ¿ g0 where one mode
suppresses the other one suRciently strongly that standing wave patterns do not form [105,417].
To understand how the dynamical behavior or sources is intimately related to that of fronts and
to the transition from the convectively unstable to the absolutely unstable regime, consider 3rst the
case in which there are no left-moving patterns, AL = 0. In this case the amplitude equation (140) for
AR reduces to the single CGL equation (137) considered in the case of the advected Taylor vortex
pattern. As we already mentioned there, fronts in the cubic CGL equation are pulled, and hence the
velocity of the rightmost front in the top panel of Fig. 40 (it is sketched with a dashed line and
connects the unstable state at x → −∞ to the saturated 3nite amplitude state for x → ∞) is simply
∗ , where v∗
s0 − vCGL
CGL is the linear spreading velocity of the single CGL equation, given in (138).
Furthermore, ca given by (139) marks the transition from the absolutely unstable regime for larger
to convectively unstable regime for smaller . Thus, for ¡ ca the right front actually recedes in
the positive x-direction, for ¿ ca the growth is strong enough that it moves upstream. See the top
panel of Fig. 40.
The picture of how sources are built from fronts, as it has emerged from theoretical studies
[102,417] is the following. Consider two widely separated fronts in AL and AR , as sketched in
Fig. 40(a). When ¿ ca the two fronts move towards each other; once they get close they form a
stationary coherent source solution which sends out waves of the same wavelength and frequency
to both sides [417]. 73 Very much like what we saw in the previous section, suRciently deep into
72
We stress here that the term “right-moving” and “left-moving” refers to the group velocity, in particular also in the
de3nition of a source. In principle, it is possible that the phase velocity of the traveling waves is opposite to the group
velocity. If this is the case, to the eye it appears that the pattern runs into the source instead of being sent out by it. See
[417] for further discussion of this.
73
We can thus think of the sources in the absolutely unstable regime ¿ ca as bound states of two pulled fronts. This
is somewhat reminiscent of the pulse solutions in the single quintic CGL equation, which can be thought of as a bound
state of two pushed front solutions [424].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
131
Fig. 40. Schematic illustration of how incoherent and coherent sources in traveling wave systems are built from front
solutions. In the upper panel, we start from two isolated front solutions which are so far apart that they are non-interacting.
In the absolutely unstable regime, for ¿ ca , the two fronts move together. The coherent source solution they then form
can be thought of as a bound state of the two fronts (lower left panel). In the convectively unstable regime, ¡ ca , the
two fronts would move apart in the absence of interactions and of noise. In practice, a wide incoherent Kuctuating source
results. Whether the average source width is determined by the noise or the interactions, is at present unknown. Possibly,
this also depends on the initial separation of the fronts.
the absolutely unstable regime ¿ ca the coherent sources give thus rise to coherent traveling wave
patterns with only very small Kuctuations.
In the convectively unstable regime ¡ ca the two fronts of Fig. 40(a) would move in3nitely
far apart if there were no Kuctuations and no interactions. The discussion of the previous section
would lead one to expect incoherent pattern dynamics in this regime, which originates from the
center of the convectively unstable region between the two fronts, as indicated in Fig. 40(c). This is
indeed what is found in numerical studies [417], but the origin of the Kuctuations is not completely
understood in this case. Clearly, if a suRciently large random noise is added to the coupled equations
(140) and (141), one will enter a similar incoherent Kuctuation-dominated regime like the one we
discussed in Section 3.11. The numerical studies give reason to believe, however, that in the small
noise limit, another intrinsic incoherent dynamical regime exists where the Kuctuations result from
the interaction of the two fronts in their tails, presumably via the generation of phase slips which
give rise to intrinsically chaotic behavior which in turn is advected away towards the front-like
regions.
One way in which the crossover between the two results shows up is in the width of the sources,
de3ned as the distance between the two points where the two amplitudes reach a 3xed fraction of
the saturated value. In Fig. 41(a) we show the results for the inverse of the (average) source width
in numerical simulations of Eqs. (140) and (142). When is decreased towards ca , the width of
the coherent source solutions increases rapidly: the width of the coherent source solutions appears
132
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 41. Results of numerical simulations of the behavior or sources in the coupled CGL equations (140) and (141) in
the absence of external noise. (a) Inverse average source width as a function of for the coupled equations with s0 = 1:5,
c1 = −1:7, c2 = 0, c3 = 0:5, g2 =g0 = 2, and 0 = $0 = 1. Note the crossover at cso just above ca = 0:14, and the fact that
the width diverges inversely proportional to for small . (b) Space–time plot of the local wavenumber of the Kuctuating
source for = 0:11 ¡ ca = 0:14, illustrating the Kuctuations of the width and the incoherent dynamics it entails in the
traveling wave domains away from it. In the black region the sum of the modulus of the two amplitudes have fallen
below 10% of the saturated value; the light and dark streaks correspond to hole-like wavenumbers packets sent out by
the source. From Pastur et al. [335].
to diverge at ca . Just before ca , however, the width becomes so large that the Kuctuation eDects
from the region where both amplitudes AR and AL are small, take over. This happens at the point
marked by cso in Fig. 41(a). For smaller values of sources are incoherent and their width scales
as −1 . Fig. 41(b) shows a space–time plot of an incoherent source in this parameter range: the core
(black region) Kuctuates in time, and these Kuctuations are reKected in the grey regions where the
amplitudes are close to their saturation values. 74 As noted before, whether the incoherent source
Kuctuations arise from numerical noise in the simulations or from intrinsic dynamics in their center,
is at present not completely clear. It is possible that there is no unique answer: when we generate
source solutions starting from initial conditions with two widely separated fronts, it is conceivable
that it depends on the initial separation of these fronts, which mechanism dominates.
Recent experiments on traveling waves near a heated wire 75 have con3rmed these theoretical
predictions [335,336]. In such experiments, a wire is suspended a few millimeters below the surface
of a Kuid, and heated with the aid of an electrical current [136,431,432]. Above some critical heating,
the temperature pattern near the wire exhibits a bifurcation to waves which travel along the wire;
by detecting the deformation of the surface, they can be detected with high precision. Measurements
of this type not only con3rm that the bifurcation is supercritical, and hence that the above coupled
CGL equations are the appropriate amplitude equations, but also yield all parameters entering formula
(139) for the crossover value [335,336]. The value obtained from such measurements is ca = 0:14 ±
0:02. In agreement with the results of the theoretical and numerical studies, a crossover in the
inverse width of the sources is found precisely at this value, see Fig. 42(a) [335,336]. Furthermore,
74
Not only is the phase of the amplitudes Kuctuating in these regions, the source also sends out coherent structures
at irregular intervals. These are visible as the light streaks, and appear to correspond to so-called homoclon solutions
[416,418].
75
In a related experiment on hydrothermal waves in a cell heated from the side, the transition from the convectively to
absolutely unstable regime was recently also veri3ed quantitatively [178].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
133
20
0
(a)
↓
40
t→
1/width (m -1 )
60
0
0.1
ε
0.2
0
x→
(b)
Fig. 42. (a) Dependence of the inverse of the average width of a source on the reduced control parameter which
measures the distance from the threshold at which traveling waves appear in the heated wire experiment of Pastur et al.
[335,336]. The value of ca which was obtained independently from measurements of the group velocity s0 , the correlation
length 0 and the time scale $0 , is indicated by an arrow. At this point, a crossover in the inverse source width as a
function of is observed in the same way as in the numerical simulations, see Fig. 41(a). (b) Space–time diagram of the
local wave number of a source in the experiments for = 0:11, i.e., in the incoherent source regime. The extent of the
x-axis is 42 cm, the total time is 10485 s.
as Fig. 42(b) illustrates, space–time traces of the dynamics in the convectively unstable regime
below ca show that the sources are wildly Kuctuating in this regime. Very much like what was
found for the width of the frequency peak in the traveling Taylor patterns in Fig. 39(a), the peak
in the spectrum rapidly broadens as is reduced below ca [336].
We 3nally stress again that just above a supercritical transition, the situation is generally rather
simple because the fronts are pulled: For this reason, the transition from convective to absolute
instability is given by the linear criterion (the sign of s0 − v∗ ). When a transition is subcritical
(inverted), the fronts near the transition are usually pushed (see Section 2.7.5), and then the distinction between the two regimes is determined by the velocity of the pushed front, for which
no general results are known. As an example of this, we may mention that a secondary bifurcation observed in a cell heated from the sides is subcritical, and that a pushed-front mediated
transition from the convectively to absolutely unstable state was recently observed in such an
experiment [179].
The discussion in this session clearly illustrates that fronts are important building blocks of the
dynamics of traveling wave systems. Although this behavior has been mapped out suRciently well
that quantitative predictions can be made in realistic cases, on a more fundamental level many issues
are poorly understood. We already mentioned that the origin of the Kuctuating sources is not understood very well—is external noise necessary or is the deterministic interaction between the tails
of the fronts suRcient to give the incoherent behavior? Can one understand some of the behavior
from studying the (non)existence of coherent source solutions following the methods of [374]? Do
sources send out homoclon solutions [418,419]? Why do sources seem to conform experimentally
to those arising in amplitude equations, while sinks do not?
134
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
3.13. Chemical and bacterial growth fronts
The issue of front propagation into unstable states often plays a role indirectly in theoretical
analysis of waves and fronts in coupled reaction–diDusion equations—e.g., fronts are an important
building block of spirals. Nevertheless, clean examples of single fronts in realistic experimental
situations do not appear to be abundant. In this section we will 3rst brieKy discuss few of the
results of a series of experiments designed speci3cally to study fronts, and then brieKy touch on
the broader implications of the diDerence between pushed and pulled fronts for pattern dynamics in
coupled reaction–diDusion systems.
In the last decade, experimentalists have been able to develop chemical reactors in which the
Turing instability and other chemical reaction patterns could be probed [76,120,152,254,263,293,327].
The Turing instability is the stationary bifurcation to periodic patterns in coupled reaction diDusion
systems, which may occur when the activator (the component with autocatalytic characteristics) has
a diDusivity which is signi3cantly smaller than that of the inhibitor [152,311,435]. The suppression
of the activator diDusion coeRcient is experimentally achieved by reversibly binding to an immobile
species attached to an aerogel; this same gel also allows the continuous feeding of the reactants
without inducing convection.
A few years ago, HorvIath et al. [201,398] and De Kepper and coworkers [110,173] have introduced
variants of such experiments that allow them to systematically produce and study fronts and the
two-dimensional patterns they give rise to. The basic reaction is a chlorite oxidation of the S4 O26−
ion. In this reaction, the hydrogen H+ ion plays the role of the autocatalyst, and its diDusion is
suppressed in a controlled way by incorporating carboxylate groups which reversibly bind it in the
polymer gels (even an electric 3eld may change the eDective diDusion ratio [399]). Initially planar
reaction fronts were created by cutting the polymer gel with the reactants into two, treating one of
the parts so as to induce a reaction, and then putting the two parts carefully back together again in
a sealed cell to prevent evaporation.
The dominant reactions in this system can to a good approximation be described by the following
dimensionless coupled reaction–diDusion equations [110,201]
9t " = ∇2 " − "J2 (> + 7") ;
(142)
9t J = +−1 ∇2 J + 6+−1 "J2 (> + 7") ;
(143)
where " is the dimensionless S4 O26− concentration and J the dimensionless H+ concentration, is
the ratio of diDusion coeRcients of these two components in the absence of the binding of the H+
to the carboxyl groups, and > is a parameter which depends on the relative chlorite excess. The
coeRcient + accounts for the reversible binding of the hydrogen: as + increases, both the eDective
hydrogen diDusion and the eDective reaction rate is suppressed.
Note that the state J = 0 (no hydrogen) in this system of equations is unstable: as soon as J
becomes nonzero the autocatalytic reaction term in these equations makes J increase. However, since
this reaction term is quadratic in J, to linear order in J there is no instability. Hence there is no
nonzero linear spreading speed v∗ , and fronts in this equation are always of the pushed type.
Theoretical studies [200,243,244] of fronts in such coupled reaction–diDusion models have shown
that when the eDective diDusion coeRcient of the autocatalytic J component is suRciently small,
the fronts exhibit a long-wavelength instability of the type found for the Kuramoto–Sivashinsky
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
135
(b)
Fig. 43. (a) Front images in the experiments by TIoth and HorvIath [398]. The upper image corresponds to the regime where
the front is stable, while the lower one to the regime where the eDective hydrogen diDusion coeRcient is so small that
the planar interface is unstable. The fronts are advancing upward. (b) Traces of the front positions in these experiments
at time intervals of 120 min in the regime corresponding to the lower image of (a). The dynamics of the interfaces in
this regime is very similar to that found in the Kuramoto–Sivashinsky equation, the generic lowest order equation just
above threshold of a long-wavelength interface instability.
equation discussed in Section 2.11.4. In agreement with this, TIoth and HorvIath [398] found that
when the hydrogen ion diDusion was suRciently suppressed, the fronts exhibited the lateral type of
structures familiar from the Kuramoto–Sivashinsky equation [243,244,277,351,381,386], in agreement
with the theoretical predictions. An example of some of their observations is shown in Fig. 43.
To what extent are these results of interest from a more general perspective? The answer lies in
the deep connection with the issue discussed later in Section 5: because the fronts in this reaction–
diDusion system are pushed, the spectrum of the linear stability operator of a planar front is gapped,
and the dynamically important region for the stability modes is the front region itself. Because
of this, in the thin interface limit the front behavior can be described by a moving boundary or
eDective interface approximation, in which the front is viewed as a line (in two dimensions) or a
sheet (in three dimensions) of vanishing width. As we shall argue below, this immediately implies
that for suRciently small ratio of diDusion coeRcients, the reaction front exhibits a long-wavelength
instability, and this in turn means that just above the instability threshold, the interface dynamics
maps onto that of the Kuramoto–Sivashinsky equation (96). 76
The point now is the following. As we will discuss in Section 5 pulled fronts cannot simply
be viewed as a moving boundary or interface in the limit when their nonlinear transition zone is
thin, because their dynamically important region is the semi-in3nite region ahead of the front. On
the other hand, when a front is pushed, the situation is very diDerent and actually much simpler:
in the limit where their width is much smaller than the typical length scale of the pattern (e.g., the
typical front radius of curvature), pushed fronts are amenable to a moving boundary or e>ective
76
The prefactor of the linear terms in the Kuramoto–Sivashinsky equation are 3xed by the coeRcients in the dispersion
relation of the weakly unstable interface mode. The prefactor of the nonlinear term is 3xed by the fact that for an isotropic
system, the projection of the velocity onto the overall propagation direction of a piece of interface of height h whose
normal makes an angle with the propagation direction, is simply vplanar =cos = vplanar (1 + 12 (∇h)2 + · · ·). For a more
explicit derivation of the Kuramoto–Sivashinsky equation and a study of the chaotic behavior in the context of the present
type of reaction–diDusion equations, see, e.g., [277].
136
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
β phase
α≈0
α phase
β≈0
enhanced α gradient:
- more reagent
- destabilizing effect
enhanced β gradient:
- more autocatalist
- stabilizing effect
reduced α gradient:
- less reagent supply
- destabilizing effect
reduced β gradient:
- less autocatalist
- stabilizing effect
Fig. 44. Schematic illustration of the origin of long-wavelength lateral instabilities of a fronts in coupled reaction diDusion
equations which describe the propagation of the domain of an “autocatalytic” substance J (e.g. bacteria in a bacterial
growth model) into a region with abundant reactant " (nutrient in the bacterial case). The front is indicated by the solid
line and is moving to the right. If it is a pushed front, its dynamics can be mapped onto an eDective interface model or
moving boundary problem for perturbations on length scales much larger than the interface width. For long-wavelength
perturbations of this eDective interface or boundary (drawn with a solid line), the supply of the reactant is enhanced at
forward protrusions and decreased for the protrusions staying behind. This eDect clearly tends to enhance the instability.
On the back side, the situation is the opposite: the diDusion of J towards the interface is reduced behind the forward
protrusions, and this tends to stabilize the interface. Which of the two eDects dominates, depends the ratio of diDusion
coeRcients. If the diDusion coeRcient of J is suRciently large relative to that of the " reactant, then the stabilizing eDect
wins, while if the diDusion coeRcient of J is small enough, the destabilizing eDect dominates and the interface is unstable.
For pulled fronts, the argument does not hold because they cannot be described in an eDective interface approximation.
interface approximation in which the front is treated as a line or sheet of zero thickness at which
the outer 3elds obey certain boundary conditions. As we illustrate in Fig. 44, a propagating reaction
diDusion front which eDectively leads to a replacement of one species ahead of it by another one on
the back, will generically exhibit a long-wavelength instability of the Kuramoto–Sivashinsky type
in the limit in which the di>usion coe?cient of the species behind the front (the autocatalyst in
the language used above) is much smaller than that of the one in front (the reactant). 77
We need to stress that the above analysis does not give a prediction concerning the lateral instability of fronts which are pulled. Indeed, the discharge fronts discussed in Section 3.5 are examples of
pulled fronts which do exhibit a long wavelength instability, while for the fronts in coupled chemical
77
For moving fronts, the long-wavelength instability that arises for suRciently small diDusion coeRcients of the phase
behind the front is intimately related to the Mullins–Sekerka instability discussed in Section 3.17. This may not be obvious
at 3rst sight, since the dispersion relation (158) is not analytic in k, while the actual dispersion relation near threshold
of the long-wavelength instability of an interface has of course an expansion in k 2 . The reason for the diDerence is that
the usual Mullins–Sekerka dispersion relation is derived in the limit that the interface growth velocity is small enough
that the diDusion equation of the outer phase ahead of the front can be approximated by the Laplace equation. Deep into
the unstable region of the phase diagram, the dispersion relation of bacterial growth models like the one speci3ed by
Eqs. (146) and (147) becomes more like the Mullins–Sekerka form (158)—see e.g. [306].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
137
reaction–diDusion equations of the form
9t " = ∇2 " − >"J − "J2 ;
(144)
9t J = ∇2 J + >"J + "J2 ;
(145)
HorvIath et al. [200] have found strong numerical indications that the fronts were always weakly
stable in the pulled regime > ¿ 12 . As soon as they were pushed for > ¡ 12 , they were found to
exhibit a long-wavelength instability, in accord with the above arguments. From a diDerent angle,
Kessler and Levine [228] have also found results which are consistent with our scenario: they pointed
out that if a set of continuum reaction–diDusion equations with pulled fronts is simulated with a
discrete lattice model, the resulting model does obey a long-wavelength lateral front instability of
the type discussed above if the ratio of diDusion coeRcients is suRciently small. This is consistent
with the observation, to be discussed in Section 7, that the eDective cutoD provided by the particles
makes the front (weakly) pushed, and hence that it should have a long-wavelength instability in the
limits in which the diDusion coeRcient of the phase on the back is much smaller than the one in
the phase ahead of the front.
The general theme of this section is also of immediate relevance for biological growth models. E.g.,
bacterial colonies can exhibit growth patterns which are reminiscent of DiDusion Limited Aggregation
clusters [292] and other growth patterns [39,40,107,183,288]. Various reaction diDusion models have
been suggested to explain some of this behavior [41,107,183,298]. In particular, it has been argued
[183,231,236] that in cases in which the bacteria secrete a Kuid which acts like a lubricant for their
motion, a model with a nonlinear diDusion coeRcient of the type
9t " = ∇2 " − "Jm ;
(146)
˜ + "Jm ;
˜ · (Jk ∇J)
9t J = ∇
(147)
with m = 1 would be appropriate. In this case, J is the dimensionless bacterial density, and " plays
the role of the nutrient 3eld. As we already mentioned above, fronts in this model are pulled in
the case k = 0, m = 1, and for any m ¿ 1 the fronts are pushed and do have a long-wavelength
lateral instability. Likewise, with nonlinear diDusion, k ¿ 0 but bilinear kinetics (m = 1), fronts are
pushed and have a long-wavelength lateral instability for small enough [306], again in accord with
our general arguments. 78 Thus, these considerations are an important ingredient for chemical and
biological model building: the above reaction–diDusion model (146), (147) with m = 1, k = 0 is a bit
of a singular case, 79 as for any m ¿ 1 or k ¿ 0 the fronts are pushed and exhibit a long wavelength
instability for small .
It is important to stress that although I believe the above scenario to be generally true on the basis
of the strength of the arguments of Section 5 and those illustrated in Fig. 44, this powerful line
of argument is, to my knowledge, relatively unexplored—I have not seen it discussed explicitly in
78
Technically, the situation for the bacterial growth model (146), (147) with m = 1 is somewhat more complicated than
we present it here: in the limit ↓ 0 the growth fronts have all the properties of an interfacial growth model with a
Mullins–Sekerka like instability, but the moving boundary approximation is probably not rigorously correct, as it is not
quite justi3ed on all physically relevant length scales [306].
79
Actually, to my knowledge, neither the singular behavior in the limit k ↓ 0 nor the one in the limit m ↓ 1 has been
worked out; these limits appear to be interesting technical challenges [306].
138
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
the physics or mathematics 80 literature, and it should be considered as an interesting line of future
research to investigate or (dis)prove this argument. The considerations are clearly an important
ingredient for chemical or biological model building.
We 3nally note that the fronts we have discussed here connect domains where the system itself
does not have a 3nite-wavelength instability. Fronts which generate a state which is unstable to a
Turing or Hopf instability, or which propagate into a state with a pattern due to this instability, are
discussed in [374].
3.14. Front or interface dynamics as a test of the order of a phase transition
When the thermodynamic phase transition between two phases is of 3rst order, both individual
phases are stable to small perturbations of the order parameter in the neighborhood of the transition.
This immediately implies that there is then a nucleation barrier for the formation of a droplet or
nucleus of one phase in the other. Since this nucleation barrier is large near the transition (as it
proportional to a power of the ratio of the 3nite surface tension of the interface between the two
phases and the diDerence in chemical potentials, which becomes arbitrarily small near the transition),
the nucleation rate for such droplets of the other phase is small. On the other hand, once an interface
exists, it can usually grow with a speed which is linear in the driving force, i.e., the diDerence in
chemical potentials or free energy WF. 81
v ∼ WF ∼ T − Tc ;
(148)
where Tc is the transition temperature. Here we used the fact that near 3rst order transition the
diDerence in free energy is linear in the temperature diDerence. The fact that the nucleation rate is
exponentially small near the transition, while interfaces respond linearly to the driving force, is the
reason for the ubiquity of interfacial growth phenomena in physics.
Near a second order transition, the situation is very diDerent. For our purposes the important point
is that the response is very asymmetric. If the system is rapidly heated up from below Tc where the
system is ordered to above Tc , the driving force for the order to disappear is 3nite everywhere in
the bulk of the system, so the order parameter relaxes homogeneously in the bulk of the system—
one normally will not see a propagation of domains of the disordered phase into ordered domains:
Instead, the order just dies away homogeneously. On the other hand, when the system is quenched
from above to below Tc , the most common situation is that patches of the stable ordered phase grow
out; usually the order parameter has a diDerent sign (say magnetization up or down in the cases of
systems with an Ising symmetry) or direction (in cases where the order parameter is a vector) in
diDerent patches, and so the initial phase is then followed by one where the domains coarsen in size
or where defects anneal out. However, if the Kuctuations are small it may happen that the ordering
80
Apparently, in some cases which we would refer to as an example of a coupled reaction–diDusion problem with pulled
fronts, it has been noted that standard analysis trying to prove convergence to a moving boundary description breaks down
(D. Hilhorst, private communication), but the connection with the general scenario advanced here appears not to have
been made.
81
An exception are faceted crystals which in the absence of dislocations grow via the nucleation of new layers on
top of the existing one. This nucleation rate is again small for small driving forces. If there are screw dislocations
at the surface, these can act as sources for step motion, and the growth rate is quadratic in the chemical potential
diDerence [42].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
139
(b)
Fig. 45. Schematic sketch of the relation between the order of a phase transition and the behavior of the interface velocity
near the critical point. The qualitative behavior is indicated within a Landau picture, but the behavior holds more generally
for non-mean-3eld systems. (a) The case of a second order phase transition. The upper part indicates the behavior of the
free energy density as a function of the order parameter below and above Tc , while the lower part illustrates that the
interface velocity is linear in the deviation of the temperature at the interface from the critical value. (b) Similarly for
the case of a second order phase transition. Note that the steady state propagation of an interface into a domain of the
ordered state is not possible, as there is a 3nite driving force on the bulk order parameter to relax to zero. On the ordered
side, thefront velocity depends nonlinearly on the deviation from the critical point. In Landau theory this dependence is
as v ∼ |Tc − T |.
in this case occurs mostly through the growth of domains of the stable phase into the disordered
phase. The velocity of these interfaces will scale as =$ where is the correlation length which
diverges as |T − Tc |−4 at a second order transition, and $ is the characteristic correlation time which
diverges as |T − Tc |−4z [199]. So we generally get a power-law scaling v ∼ |T − Tc |4(z−1) . In mean
3eld theory, with 4 = 12 and z = 2 for diDusive systems, this gives v ∼ |T − Tc |1=2 .
These considerations can be illustrated simply in a Landau mean 3eld picture for an order parameter described by the dynamical equation
F
9
˜ 2 + F()] :
=−
; with F = dr[ 12 (∇)
(149)
9t
The dynamics implied by this equation is such that the free energy F is non-increasing in time
(F acts like a Lyapunov function, see footnote 58), so it tends to drive to the value at which
the free energy density F() is minimal.
In the case of a 3rst-order transition, the function F() has the form sketched in Fig. 45(a): there
are two local minima (or three if the order parameter is symmetric under a change of sign). At high
temperatures, the one corresponding to the disordered phase ( = 0) is the absolute minimum, but
as the temperature is lowered, the other one corresponding to the ordered phase comes down, and
at T = Tc the value of F at the two minima is the same. Below Tc , the latter one is the absolute
minimum. If one consider a planar interface solution = (-) in a frame - = x − vt moving with
140
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
velocity v in this case, Eq. (148) reduces to
−v
F
9
=−
;
9
which upon multiplying with 9- gives
∞
d- 9- F= F((∞)) − F((−∞))
∞
∞
=
:
v = −∞
2
2
−∞ d-(9- )
−∞ d-(9- )
(150)
(151)
The term in the numerator is a unique number for functions F() like those sketched in Fig. 45(a).
For stationary interfaces, when the minima of the free energy in the two phases is the same, this
term is related to the excess energy associated with the presence of the interface [56,192,428]: we
can then interpret it as the surface tension. Since this term is positive, the above expression con3rms
that the interface velocity is linear in the diDerence in the free energy densities of the two phases it
separates, and hence linear in the temperature, as sketched in the 3gure and anticipated already in
Eq. (148). Moreover, it propagates in such a direction that the domain with the lowest free energy
density expands.
The function F() corresponding to a second order transition is sketched in Fig. 45(b), and the
prototypical dynamical equation for this case is of course the F-KPP equation (1) with f(u)=u−u3
and ∼ T − Tc . For ¿ 0, i.e., below T√
c , we have indeed the possibility of fronts propagating into
the unstable state u=0 with speed v∗ =2 . As was already stated above, for ¡ 0, (above Tc ), any
nonzero initial condition for u is driven rapidly to zero, and hence no front propagation is possible.
Thus, as illustrated in the lower part of Fig. 45(b), near a second order transition, propagation of
fronts is very asymmetric relative to Tc , on one side fronts or interfaces are possible, but their growth
velocity is a nontrivial power law of the distance from criticality, T − Tc , while on the other side
interface motion does not occur.
Normally, one does not study the motion of an interface in order to determine the order of a
phase transition, of course. However, these considerations have been of use in at least one case, the
study of the so-called Halperin–Lubensky–Ma [194] eDect near a nematic–smectic A transition. In
the nematic phase, the orientation of the long liquid crystal molecules acquire a directional order:
on average they all point in the same direction [114]. This directional order is the slow mode of a
nematic phase that is responsible for much of the special behavior and applications of these materials.
In the smectic phase, the molecules also obtain a layered ordering, and the smectic A phase is the
one where the molecules point on average along the normal to these layers. In a classical Landau
theory, the nematic to smectic-A transition can be both of 3rst and of second order: if we think
of in the discussion above as the smectic layering order parameter, then in a Landau theory,
the free energy density F() can have either of the forms sketched in Fig. 45. However, Halperin
et al. [194] showed that when the coupling to the director Kuctuations is taken into account, and
when subsequently the director Kuctuations are integrated out, 82 the Landau expression acquires
a cubic term ||3 which renders the transition weakly 3rst order in the regime where without
this term it would be of second order. The Halperin–Lubensky–Ma eDect is thus an example of
a Kuctuation-driven 3rst order transition.
82
The analysis of Halperin et al. [194] also applies to type I superconductors: the coupling to the gauge Kuctuations
can drive the normal to superconductor transition weakly 3rst order in some regimes.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
141
The 3rst dynamical indications for the existence of this Kuctuation eDect came from a series of
experiments near the nematic to smectic-A transition of Cladis et al. [87]. When the temperature was
changed through Tc for a series of liquid crystals mixtures which before had been concluded to span
a tricritical point where the transition changed from 3rst order to second order, propagating interfaces
were observed, both upon raising the temperature and upon lowering the temperature through Tc . As
explained above, this might be taken as a sign that the transition was in fact always at least weakly
3rst order for all compositions. 83 A more careful analysis by Anisimov et al. [8] later con3rmed
this and showed the consistency of these dynamical measurements with measurements of the latent
heat and the correlation length near the transition [7]. These authors took the Landau free energy
expression
˜ 2 + a2 + b||3 + c4 + d6 ] ;
F() = dr [ 12 20 (∇)
(152)
with the cubic term motivated by the analysis of Halperin et al. which as mentioned above always
renders the transition 3rst order. They then 3tted the parameters a and c in this expression to both
the latent heat data and the X -ray measurements of the correlation length of this series (for b = 0,
the point where c = 0 marks the tricritical point). The other parameters were 3xed once for the
whole series of mixtures. The 3t of the measurements of the cross-over function for the latent heat
(which is always nonzero for b = 0) is shown in Fig. 46(a). Once these coeRcients are determined,
one can then calculate the slope v= ∼ v=|T − Tc of the interface response near the transition, apart
from an overall factor that sets the scale. As Fig. 46(b) shows, the curve obtained this way from
static measurements 3ts the experimental data for the interface mobility v= remarkably well for the
9CB–10CB mixtures. Taken as a whole, the dynamical measurements together with the theoretical
analysis thus give quite strong evidence for the prediction by Halperin et al. [194] that the coupling
to the director Kuctuations drive the nematic to smectic-A transition weakly 3rst order.
3.15. Switching fronts in smectic C ∗ liquid crystals
In the previous section, we already encountered the smectic-A phase of a liquid crystal. In this
phase the molecules form layers. Along the layers the molecules are Kuid-like (no ordering), but
the orientation of the molecules is aligned on average along the normal of the layer. In a smectic-C
phase, the molecules again have a layered ordering, but on average they are tilted in each layer with
a 3xed angle relative to the normal. The projection of this tilt onto the layers forms an azimuthal
angle, which is an important slow hydrodynamic variable for the smectic-C phases. In the smectic-C∗
phase, 3nally, this angle rotates over a small angle from layer to layer, so that it makes a full twist
of 2 over a mesoscopic distance, the “pitch” of the liquid crystal. This situation is sketched in
Fig. 47(a). In this 3gure, the local polarization (which is normal to the average orientation of the
83
The tricky part experimentally is to rule out the possibility that there were no temperature gradients in the experiments
which induced an eDective interface—after all, if the temperature is above Tc on one side of the sample and below Tc on
the other side, one will also create an interface if the transition at constant temperature is second order. Care was taken
in the experiments [8,87] to rule out such gradients, but the strongest evidence for the Kuctuation-induced weakly 3rst
order transition actually comes from the consistency of the dynamical measurements with the latent heat measurements
and the X -ray measurements of the correlation length [8].
142
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 46. (a) Normalized universal scaling form for the change of entropy WS at the transition (which is proportional to the
latent heat) in a three series of liquid crystal mixtures. The full line is a 3t to a scaling form obtained from the Landau
free energy expression (152). The variable along the horizontal axis is a scaled composition variable. In the absence of
the cubic term proportional to b in Eq. (152), the curve would vanish linearly at y − y∗ = 0, and would be zero to the
left of it: y∗ marks the tricritical point in the scaled parameters and in the absence of the cubic term. From Anisimov
et al. [8]. (b) The measured interface mobility v= plotted as a function of the concentration in the 9CB-10CB mixtures
according to the analysis by Anisimov et al. [8]. The full line is obtained from the Landau expression obtained by 3tting
the parameters to latent heat data like in (a) and to measurements of the correlation length, apart from an overall factor
that sets the scale.
molecules) is indicated with little arrows, and the angle this vector makes with the x-axis is denoted
by ; the z-axis is the coordinate along the normal of planes.
What makes the smectic-C∗ phase especially interesting from our perspective is that the angle can be oriented both with an electric 3eld and with a magnetic 3eld. Without a 3eld, the twist of
the director is uniform, i.e. = 2z=p, where p is the pitch length, which depending on the material
can range from a few tenth of a micron to several microns. If we consider for simplicity the case
that the angle is uniform within each layers, then in the presence of electric and magnetic 3elds
perpendicular to the layers, the free energy per unit area can be written as [88,114,274]
2
K 9
WE 2 WMH 2
F = dz
− q0 − PE cos +
+
cos2 :
(153)
2 9z
8
2
Here P is the ferroelectric polarization, which points normal to the direction in which the molecules
are tilted. The second term describes the dielectric and diamagnetic coupling: W is the dielectric
anisotropy and WM similarly the diamagnetic anisotropy [114]. Note that the electric and magnetic
3eld enter the same way. In the theoretical analysis below, we therefore simply put H = 0 and
consider the two cases in which W is positive or negative. Experimentally, however, the fact that
there are two diDerent contributions to this term is important. First of all, it allows one to shift the
importance of this term relative to the polarization term. Secondly, if W and WM have opposite
signs it opens up the possibility to change the sign of this term. As we shall see below, this has
important implications for the front dynamics.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
143
(b)
Fig. 47. (a) Sketch of a smectic-C∗ phase of a liquid crystal. In this phase the molecules are layered, but their average
orientation is tilted relative to the normal to the planes. The local polarization Ps , which is normal to the c-director, is
indicated with arrows in this 3gure. From layer to layer, this angle makes a small rotation, so that over a large distance
the angle is wound over 2. From Maclennan et al. [274]. (b) Schematic illustration of the stable angle solution (z) of a
smectic-C∗ in a strong electric 3eld E along pointing in the positive x-direction (full line) and in the negative x-direction
(dashed line). Note that in the 3rst case, there are wide plateaus at 0; ±2; ±4; : : :, while in the latter case these are at
±; ±3; : : : . Upon switching the 3eld direction, each domain wall where changes rapidly before the switching splits
into two fronts propagating into the two adjacent plateaus, which have been made unstable due to the reversal of the
3eld. From [427].
Suppose we start from a case without any 3elds, so that the smectic is in the uniform helical state
= 2z=p, and switch on the electric 3eld to a large positive value. Then the free energy density
is lowest for ≈ 0; ±2; ±4, etc. Because of the twist in the initial state, the smectic will form
a series of domains of length close to the pitch length p, separated by thin domain walls where rapidly changes by 2. This situation is sketched in Fig. 47(b) with the full line. If the 3eld is now
rapidly reversed, E → −E, then the free energy density is lowest in the regions where = ±; ±3,
etc. In the limit in which the 3elds are large so that the domain walls are thin, the original walls
will then split into two fronts which propagate into the domains which have been made high energy
domains by the 3eld reversal.
In the absence of the dielectric and diamagnetic terms the situation is very simple: the switching
of the 3elds makes the state in the domains unstable, since the prefactor of the sin term in the free
energy changes sign. Hence in the high 3eld limit we have a clear case of pulled front propagation
into unstable states. The general situation is more intricate as it depends on the sign of the dielectric
and diamagnetic term; let us de3ne for H = 0
W|E|
(H = 0) :
(154)
4P
If " is negative, then the dielectric 3eld contribution has extrema for the same angles where the
polarization term is extremal, while when " is negative, it is minimal for = ±=2; 3=2, etc.
The full behavior of the 3eld contributions to the free energy density as a function of are
sketched in Fig. 48(a) for various values ", both for 3eld E ¡ 0 (upper panel) and for 3eld E ¿ 0
"=
144
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 48. (a) Electrical free energy density (153) of the smectic-C∗ as for various values of ". The upper panel corresponds
to the case E ¡ 0 and the lower panel to the same 3eld positive 3eld strength. From Maclennan et al. [274]. (b) Front
velocities for the generalization of Eq. (155) to the case H = 0 for three values of the 3eld. The full curve corresponds to
the case H = 0 discussed in the text. The crossover from pulled to pushed fronts occurs at E = 1=2 in this case. Symbols
indicate the velocity obtained numerically from simulations in runs of 3nite length. Note that the data points tend to lie
slightly below the asymptotic velocities in the small 3eld range, due to the slow power law convergence to the pulled
front speed v∗ . From [427].
(lower panel). Consider 3rst the case " ¡ 0, and take initially E ¿ 0. As the lower panel shows,
the state with = 0 then has the lowest free energy, so in the domains we have ≈ 0 (modulus
2). Now, upon switching the electric 3eld, the free energy becomes the one of the upper panel.
As the curves indicate, the state in the domains is then unstable for −1 ¡ " ¡ 0 and metastable for
" ¡ − 1. Thus, by increasing the 3eld one can continuously go from a case of front propagation
into unstable states to a case of a front propagating into a metastable state, in this regime.
Clearly, the regime " ¿ 0 is diDerent, as Fig. 48(a) illustrates. While upon increasing the 3eld E
for negative values of " the free energy density sharpens at minima, upon increasing the 3eld for
" ¿ 0 the minimum initially Kattens and then turns into a local maximum. Moreover, upon switching
the electric 3eld for " ¿ 1 the state which was stable before switching experiences a 3nite driving
force to the new minimum. This implies that in this regime one will not observe front propagation.
For smectic-C∗ liquid crystals, the moment of inertia of the molecules is normally small enough
that the inertial terms can be neglected in comparison with the viscous torque. In this regime, the
dynamical equation for the angle then becomes ;9t = −F=, where ; is a twist viscosity.
In appropriate dimensionless variables this gives for " ¡ 0 and E ¿ 0 [88,114,274]
9t = 92z + f()
with f() = E sin − E 2 sin cos ;
(155)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
145
which is nothing but the prototype equation for front propagation, the F-KPP equation (1); the only
diDerence with the usual case is that the function f() is now a periodic function of , which
allows for stable arrays of kinks 84 like those sketched in Fig. 47(b).
For a full study of the dynamics exhibited by this equation upon reversal of the 3eld direction
we refer to the work by Maclennan, Clark and Handschy [274]; we focus our discussion here on
the front dynamics which is relevant in the parameter range where the front width is much smaller
than the pitch p.
The surprising feature in this case is that the selected front velocity can be obtained analytically
both in the pulled and in the pushed regime, because it turns out that it is possible to solve for the
pushed front solutions by making the Ansatz h=E sin() in the “reduction of order method” discussed
brieKy in the example at the end of Section 2.7.1. Indeed, for the case H = 0 this yields pushed
solutions for E ¿ 1=2 with velocity v† = 1 of (155), whose analytic form is given by [88,387,427]
(156)
† () = 2 arctan[exp(− |"|)] :
√
Since v∗ = 2 E − E 2 , one indeed immediately sees that v† ¿ v∗ for E ¿ 1=2. As mentioned earlier,
the results for H = 0 can all be obtained by appropriate transformations that translates this case
back to the case H = 0 summarized above. Nevertheless, from a practical point the possibility to
play with both E and H may be quite important, since when H = 0 the front solutions are pushed
both for small and for large 3elds E. Fig. 48(b) shows the selected front speed as a function of E
for three values of the magnetic 3eld H .
As is well known, liquid crystals are important for displays; if one would want to use the present
switching eDect in applications, the switching time is of course an important parameter, and this is
inversely proportional to the front speed. For a discussion of the transient behavior as well as of the
comparison with experiments, we refer to the review by Maclennan et al. [274].
3.16. Transient patterns in structural phase transitions in solids
One possible way in which a structural phase transition in a solid can occur is when certain
atoms or ions which have two diDerent competing sub-lattices available, order below some critical
temperature. Often, such ordering phenomena in solids are strongly coupled to strain deformations
and lower the symmetry of the crystal structure. About a decade ago, it was conjectured that certain
transient metastable tweed patterns that are sometimes found near structural phase transitions might
be due to the propagation of a pattern forming front into an unstable state [373].
For the transient patterns of interest—e.g. tweed patterns associated with vacancy ordering in
YBCO superconductors or Al,Si ordering in Na feldspar—the equilibrium ordered state is to have a
homogeneous phase in which the ions are (partially) ordered on one of two available sub-lattices.
In practice, however, domains with alternating order are sometimes observed upon shock heating.
Salje [373] has suggested that these pattern arise from a pattern forming front propagating into the
structurally unstable homogeneous disordered phase. In particular, if denotes the dimensionless
kinetic order parameter with = ±1 indicating the two possible sub-lattice ordered states, and
= 0 the disordered state in a coarse-grained description, Salje [373,406] arrived, on the basis of
84
In the usual case, the equation also admits solutions which correspond to a periodic array of kinks where switches
between the stable states of f(), but these multiple kink arrays are then unstable to pairing.
146
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
a treatment of the kinetics which approximately includes nonlocal strain eDects, at the following
dynamical equation for the order parameter:
9t = 92x − 294x + 1 (9x )2 + 2 2 92x :
(157)
Note that this equation can be viewed as an extension of the EFK equation discussed in Section
2.11.1. Like that one, the stable lowest energy states are clearly the homogeneously ordered states
=±1. Moreover, since the terms linear in are exactly those of the EFK equation, it follows from
the results of Section 2.11.1 that indeed for 2 ¿ 1=12 a pulled front propagating into the unstable
disordered = 0 state will give rise to transient patterns. Simulations in [373] also con3rmed this. 85
To our knowledge, there is at present no direct evidence in support of the intriguing conjecture
that the transient modulated order patterns are caused by pattern forming pulled fronts—the diRculty
with solid ordering phenomena is that it is often hard to rule out other possible mechanisms.
3.17. Spreading of the Mullins–Sekerka instability along a growing interface and the origin of
side-branching
When the growth of an interface is limited mainly by how fast the material necessary for growth
can be transported towards it from the phase into which it grows, or by how fast heat produced
at the interface can be transported away through diDusion into the phase into which it grows, then
a growing planar interface of this type is unstable: A small protrusion of the interface towards the
phase into which it grows leads to an enhancement of the gradients in front of the interface. This
enhanced gradient leads to an enhancement of the diDusion and hence to an enhanced growth: the
protrusion will grow larger and larger, and render the interface unstable. This so-called Mullins–
Sekerka instability [73,220,247,308,309], which we essentially already encountered in Section 3.13,
underlies an enormous variety of diDusion-limited growth processes, ranging from crystal growth
from the melt or electric discharge patterns like the streamers of Section 3.5, to fractal growth
phenomena like DiDusion Limited Aggregation [22,292].
We focus our discussion on the Mullins–Sekerka instability of a planar crystal interface growing
into an undercooled melt. In this case, the dispersion relation of small perturbation in the height h
of the interface of the form h ∼ e−i!t+ikx is [73,247,308]
! = ivn |k|(1 − d0 ‘D k 2 ) :
(158)
Here vn is the normal velocity of the planar interface, as sketched in Fig. 49, d0 is the capillary
length which is proportional to the surface tension of the interface, and ‘D is the diDusion length
on the liquid side of the interface.
The linear spreading velocity associated with this dispersion relation is easy to determine; one
simply 3nds
√
∗
vMS
= 3vn ;
(159)
85
Note that the term proportional to in (157) tends to enhance the growth rate, and that it is quadratic in the amplitude.
We therefore would expect a transition to pushed fronts for suRciently large ; to our knowledge, this possibility has not
been explored, however.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
147
Fig. 49. Schematic illustration of the spreading of a localized perturbation along a growing crystal melt interface which is
unstable as a result of the Mullins–Sekerka instability. The dashed line indicates the unperturbed planar interface, which is
growing in the vertical direction with a normal velocity vn . The dashed arrows indicate the propagation of the perturbation
along the interface. In accord with the discussion of Section 2.6, the asymptotic Mullins–Sekerka front speed is larger
∗
than or equal to the linear spreading velocity vMS
given in Eq. (159).
with
∗ ≡ ki∗ = √
1
;
6d0 ‘D
kr∗ = √
1
;
2d0 ‘D
(160)
where, as before, kr∗ =Re k ∗ and ki∗ =Im k ∗ . To our knowledge, the spreading of the Mullins–Sekerka
instability along the interface has not been studied directly, neither theoretically nor experimentally. 86
We therefore do not know whether the propagation of the Mullins–Sekerka along the interface
corresponds to a pushed or pulled front, although since we know that dendritic growth is such a
strong instability without saturation, we might intuitively expect it to be pushed in the case of a
Kat interface illustrated in Fig. 49. Thus, in line with the arguments of Section 2.6, we simply
∗ . Note that if the
conclude that the front will propagate sideways with asymptotic velocity v ¿ vMS
front would happen to be pulled, the relation (160) shows that the amplitude of the oscillations
∗
grows exponentially by a factor er over one wavelength 2=kr∗ , with
r∗ ≡
2ki∗
2
= √ ≈ 3:62 :
∗
kr
3
(161)
As was also discussed in Section 2.7.6, even if a front is pulled for suRciently localized initial
∗
conditions, it can move faster than v∗ if the initial conditions are falling oD slower than e− x in
space (this case was referred to as “leading edge dominated dynamics” in Section 2.7.6 and in
[144]). For pattern forming fronts like these, it was argued in [421] that for a given spatial decay
rate = ki ¡ ∗ , the wavenumber would be the one maximizing the growth rate Im !, i.e., that kr
would be determined implicitly by the condition 9!i (k)=9kr = 0. In the present case this gives
kr2 = ki2 +
86
1
;
3d0 ‘D
(162)
The experimental diRculty is that it is very hard to prepare an unstable interface in this case; if one starts with a
stable interface and then tries to bring the interface to the unstable regime, the buildup of the diDusion boundary layer
usually gives rise to long transients. Often, the instability already arises during this transient regime.
148
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
(b)
Fig. 50. (a) Snapshot of a NH4 Br dendrite in the experiments by Dougherty et al. [133,134]. (b) Sketch of a dendrite tip
identifying the various quantities discussed in the text.
and for the velocity v of the leading edge of such pro3les this gives
!i
kr 2d0 ‘D
2
= vn
v≡
= vn
ki
ki
r
3r(1 − r 2 =(42 ))
√
√
∗
¿ vMS
= vn 3 (for r ≡ 2ki =kr ¡ r ∗ = 2= 3) :
(163)
Let us now explore a simple implication of this idea for dendrites. Dendrites are the tree-like
growth structures of the type sketched in Fig. 50(a) which the Mullins–Sekerka instability gives rise
to. Understanding the shape and velocity selection of a dendritic tip was one of the outstanding
problems in the 3eld of interfacial pattern selection in the 1980s. As discussed in various reviews
[39,40,73,226,251,338,353], it is now accepted by most workers that the size and velocity of the
near-parabolic tip are determined by the nonlinear eigenvalue problem for a uniformly growing
near-parabolic tip, and that although surface tension eDects are typically small, they act as a singular
perturbation. The picture that has emerged is thus that the occurrence of “sidebranches” on these
dendrite tips is not important for the dynamical mechanism that “selects” the tip shape and velocity.
The accepted view in the 3eld is that the sidebranches occur due to selective ampli3cation of noise
from the tip: small perturbations and Kuctuations occur at the tip of the dendrite; these grow out,
while being convected away in a frame moving with the tip itself. These ideas have been put forward
on the basis of WKB-type analysis of the spreading, growth and advection of side-branch “wave
packets” which start near the tip [23,346,347]. Though there is some evidence for this behavior
[359], experiments on this issue have turned out to be hard in general; moreover, it is not clear
how realistic a WKB-type calculation is for the experimentally relevant regime where tip radius and
sidebranch spacing are comparable. However, our analysis above gives us a very simple diDerent way
to verify this picture from the data, and in addition suggests a useful way to analyze experimental
data on sidebranches. The discussion is a slight reformulation of [425].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
149
The above picture that sidebranches arise from the ampli3cation of noise near the tip is based
on the idea that the Kanks of the smooth needle solution underlying the dendritic tip region are
unstable to the Mullins–Sekerka instability, but that the instability is convective there. Although the
sidebranches do not form a coherent front due to the Kuctuations, their envelope must propagate on
average with a velocity whose projection vsb
along the growth direction is equal to the tip velocity
vtip . For a region on the side illustrated in Fig. 50(b) where the underlying tip pro3le makes an
angle with the growth direction, this velocity can be expressed in terms of the velocity vsb of the
sidebranch front along the interface and the normal velocity:
vtip = vsb
= vsb sin + vn cos = vsb sin + vtip cos2 ;
(164)
where we used the fact that vn = vtip cos . Equating the two sides gives simply
vsb = vtip sin :
(165)
This equation immediately allows us to compare measured quantities and infer the underlying dynamics from it. Indeed, if we take the interface on the sides as locally planar by ignoring the
curvature of the underlying needle solution in this region and assume that the average wavenumber
of the incoherent sidebranches can be associated with kr above, vsb is nothing but the velocity given
∗ , the sidebranch instability is indeed convecin (163). If vsb is indeed signi3cantly bigger than vMS
tive, √
and the √selective ampli3cation of noise scenario is corroborated by this analysis too. Since
∗
v = 3vn = 3vtip cos , in our lowest order approximation we may conclude:


¿ 60o ;


sidebranch instability convective
(166)
⇒ r ¡ 3:62 ;

sidebranches = ampli3ed tip noise

 v ¿ v∗ [v obeys (163)] :
sb
sb
To illustrate this, let us apply the above line of argument to the dendrite shown in Fig. 50(a).
First of all, the angle on the Kanks of the tip region where the sidebranch amplitude is small, is
bigger than 60o . Furthermore, the amplitude of the sidebranch amplitude in this experiment was
measured explicitly by Dougherty et al. [133,134]. As Fig. 51 shows, this amplitude initially does
rise exponentially; the spatial exponential growth rate ki is easily estimated to be about 1=(16 m)
from these data. This is an indication that indeed in this case, nonlinearities lead to saturation
of the sidebranch instability, and hence that if there were no Kuctuations, the sidebranch front
would correspond to a pulled front. Since the wavelength is about 13 m in this experiment, we
3nd r ≈ 0:8 ¡ r ∗ . In addition, the sidebranch velocity vsb is also found to obey (163) to a good
approximation, which indicates that the consistency of this line of argument. Hence, both the angle
and the spatial growth rate con3rm that the sidebranches in this experiment are consistent with the
scenario [23,346] that they emerge from the ampli3cation and convection of small Kuctuations near
the tip.
Many dendrites appear very much like the ones shown in Fig. 50(a) at 3rst sight. It therefore does
appear that in most cases the sidebranches are consistent with the above ampli3ed tip-noise scenario.
Unfortunately, the dimensionless growth rate r is hardly ever mentioned in experiments, but judging
by the eye this quantity does appear to vary signi3cantly from experiment to experiment.
An interesting question remains whether in some cases, the sidebranch instability might become
∗
absolute, in that the projection of the linear spreading velocity vMS
becomes comparable to the
150
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 51. Measured root mean square amplitude (squares) of the sidebranches of the dendrite shown in Fig. 50(a) as a
function of the distance z from the tip, in the experiments by Dougherty et al. [133]. The crosses give a measure of the
amount of coarsening of the sidebranches; they are based on the amount of spectral power at low frequencies [133].
tip velocity (such a scenario was also envisioned in [248,287]). One would expect that tip and
sidebranches then become strongly coupled, and possibly that the tip region would show appreciable
oscillations. That might happen in some parameter range was already suggested by simulations of
the so-called Boundary Layer Model by Pieters [346]. Recently, Sakaguchi and Tokunaga [372]
observed such behavior in phase 3eld model calculations but the data were not correlated with the
parameter r introduced above. A good way to classify sidebranch regimes experimentally and to
search for this possibility is to measure the dimensionless growth factor r—if this value is found to
increase towards r ∗ ≈ 3:6 then the possibility of such a regime becomes more likely. Of course, for
a realistic comparison with experiments, eDects of interface kinetics and crystalline anisotropy may
have to be included, but at the level of the approximations discussed here this poses no technical
problems.
The above considerations concerning the absolute or convective nature of the sidebranch instability
in my view help us understand how two competing theories might actually emerge as diDerent limits
within a more general framework. The usual “solvability theory” which focusses on the existence
of featureless needle solutions, amounts, within a WKB approximation, to the requirement that a
smooth tip solution without sidebranch-type modes on the Kanks is stable [45]. In his “Interfacial
Wave Theory”, on the other hand, Xu [444] develops an approach in which he allows, essentially
within the same WKB approximation, a tip mode to match to divergent sidebranch modes on the
sides. 87 To my knowledge, the stability of these solutions, constructed this way, has not been
fully investigated. Based on the above analysis, I would expect such dendrite-type solutions to be
actually unstable in substantial parts of the parameter range—if so, they would presumably, in the
parameter range where this happens, give way to needle-type solutions which are convectively stable
in a deterministic approach. In other words, in the range where this happens the usual solvability
87
The turning point in Xu’s WKB analysis becomes, in some limit, the linear spreading point of our analysis, and
in principle the “interfacial wave theory” predicts the value of the parameter r introduced above. Also note that the
diDerences and similarities between Xu’s approach and the usual solvability theory are brought out most clearly if one
thinks of solvability theory in the spirit of the formulation of [45].
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
151
scenario would hold, both for the selection mechanism and for the ampli3cation of noise. In other
parts of parameter space (roughly identi3ed by generalizations of the ideas presented in this section),
needle solutions would be absolutely unstable to sidebranch modes. I would expect Xu’s picture to
emerge naturally in this regime.
Unfortunately, the dendrite problem is a hard one, and at present we have to consider the above
scenario as speculative. In addition, the reader should keep in mind that the above view on the
dendrite problem is a minority viewpoint—most workers consider the various theories as mutually
exclusive, rather than as two extreme limits of a more general framework.
3.18. Combustion fronts and fronts in periodic or turbulent media
The classic work of Landau [245], who analyzed the stability of Kame fronts in terms of a Kame
sheet of in3nitesimal thickness, separating unburned and burned gases, provides a very early example
of the clever use of a moving boundary approximation for a problem which more fundamentally is
formulated in terms of continuity equations for the temperature and composition of a combustible
mixture. The justi3cation for this approximation was later shown to be that the activation energy of
the relevant reactions is normally very high, so that the reaction rate is a very steep function of the
temperature [67,90,209,439,449]. Inherently, realistic Kame fronts are therefore in practice almost
always examples of pushed fronts: the reaction rate is very strongly (essentially exponentially)
suppressed ahead of the Kame sheet. In other words, the linear spreading speed is essentially zero.
Although the minimal model for Kames amounts to two coupled partial diDerential equations for
the temperature and composition, simple model equations have always played an important role in
combustion too. We mention two interesting examples coming from the combustion literature. In
fact, an early example of the F-KPP equation (1) with a nonlinearity f(u) which vanishes identically
in a 3nite interval (e.g. f(u) = 0 for u ¡ uc , f(u) = (u − uc )(1 − u) for u ¿ uc ) is due to Gel’fand
[180]. This form was motivated by the above observation that the reaction rate drops so fast with
decreasing temperature that it is virtually zero below some critical value.
Two other interesting classes of problems have also emerged from combustion theory: propagation
of fronts in periodic media or turbulent media. We 3rst discuss the case of fronts in periodic media.
The simple model problem which has often been studied in this context is the following extension
of the F-KPP equation [47,48]
9t u = 92x u + q(x)9x u + f(u; x) ;
(167)
and generalization to higher dimensions. In this equation, the advection rate q(x) and nonlinear
growth rate f are periodic functions of x with period L: q(x + L) = q(x), f(u; x + L) = f(u; x).
Two classes of growth functions f have been studied, those analogous to the one by Gel’fand
mentioned above, and those which give rise to pulled fronts in the F-KPP equation [e.g. f(u) =
a(x)u(1 − u)]. The latter type of problem of a pulled front in a periodic medium is especially
interesting from our perspective: It should be possible to obtain the asymptotic propagation speed
v∗ of such pulled fronts by extending our general approach using Floquet–Bloch theory for the
linearized equation. To be speci3c, consider the linearized version of (167) with f(u; x) given as
above,
9t u = 92x u + q(x)9x u + a(x)u ;
(168)
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
where q(x) and a(x) are L-periodic,
q(x + L) = q(x);
a(x + L) = a(x) :
(169)
According to Bloch’s theorem in the language of a physicist or Floquet theory in the more general
setting, the generalization of the Fourier transform become the “Bloch waves”
ũ(k; t) = e−i!(k)t+ikx U (x) ;
(170)
where U (x) is a periodic function of x, U (x + L) = U (x). Just as we did in Section 2.1 for the
Fourier transform, once the dispersion relation !(k) is obtained, one can analytically continue k into
the complex plane. This immediately leads to the conclusion that the linear spreading velocity v∗
is again given by the same saddle-point equations (12)—the periodicity of the medium is simply
encoded in the dispersion equation !(k). For solid state physicists this is no surprise: !(k) is the
analog of the band energy (k) of Bloch electrons, and as they know, the behavior of free electrons
in a solid is completely determined by the band structure.
To the best of my knowledge, the above line of analysis has unfortunately not been tried yet; we
do hope such an approach will be explored in the near future, as it would probably be the most
direct route to obtaining the asymptotic speed of pulled fronts and as interesting new phenomena
might arise from the band structure. In particular, for the simple case a(x) = 1 + a0 cos(2x=L) the
linear equation reduces to the Mathieu equation, for which it should be possible to obtain a number
of (semi-)analytic results.
Another class of front problems which has emerged from combustion theory is the propagation of a front in a spatially and/or temporally random medium as a model for turbulent combustion [225,241,341,443]. From an applied point of view, an important question is to understand
the enhancement of the combustion rate due to turbulent advection. One way in which this issue
has been approached recently within the context of model problems is to take the function f in
Eq. (167) x-independent but the advection 3eld q in this equation a space- and time-dependent random variable. This makes the 3eld u into a stochastic variable as well, and recently several workers
[1,2,153,171,172,225,400] have derived results for various probability distributions of the advected
u variable. One reason that progress can be made on this complicated problem is that for a pulled
front, the linear u-equation captures the essential elements of front propagation (more mathematically: bounds can be derived for the nonlinear equation, using the linearized advected dynamics). It
was recently also found that if the q-variable is a stochastic Levy process, the front propagation can
change drastically [281]—even an exponentially increasing speed is possible. 88
There is one important issue concerning fronts in turbulent or random media that to our knowledge has not been discussed explicitly in the literature. Real combustion fronts are pushed fronts,
because the combustion rate decreases very rapidly with decreasing temperature. In the limit of large
activation energies, Kame fronts are very thin and they can be analyzed with a moving boundary
approximation [341,439]—this amounts to the approximation in which the Kame sheet is treated as
an interface of zero thickness. As we discuss in Section 5, for pushed fronts this approximation
is indeed justi3ed, but for a pulled fronts it is not: For these, the dynamically important region is
the semi-in3nite domain ahead of the nonlinear front region. It is therefore conceivable that simple
88
We speculate that this may be related to the fact that an F-KPP equation with power law initial conditions can give
rise to an in3nite speed [256], and that the probability distribution function of a Levy process has power law tails.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
153
models like (167), whose dynamically important fronts are pulled, are not necessarily good models
for turbulent combustion. Probability distribution functions in simple combustion models might well
be very diDerent depending on whether fronts are pushed or pulled.
3.19. Biological invasion problems and time delay equations
As mentioned in the introduction, the 3rst studies of front propagation into an unstable state
were done in the context of population dynamics and biological invasion problems [163,234]. Not
surprisingly, this has therefore remained an active 3eld of research within mathematical biology. The
main focus of a large fraction of the literature is still on proving existence, stability and uniqueness
of traveling wave solutions in population dynamics models which can be considered as extensions
of the F-KPP equation. We refer to Section 14 of the books by Britton [63] and Murray [311]
or to the recent review of Metz et al. [296] for an introduction and overview of this sub3eld.
The book by Shigesada and Kawasaki [385] not only gives a good review of the theory, but also
discusses in detail a number of applications of the theory to practical invasion problems. Pulled front
propagation into unstable states plays a dominant role in a recent model of infection in the Hantavirus
epidemics [3].
Also in the bio-mathematical literature the slow power law convergence of pulled fronts to their
asymptotic speed has been discussed [296], but I am not aware of examples in that 3eld where this
phenomenon has played a signi3cant role. In real life, the fact that the entities in invasion problems
are normally discrete may play a more important role—as we shall discuss in Section 7.1, the cutoD
in the growth function that the discrete nature of the spreading population gives rise to, alters a
pulled front in a continuum equation to a (weakly) pushed front in a model for discrete variables.
A recent development in this 3eld, reviewed recently by Fort and MendIez [165] has been to
consider the eDects of time-delay on a front [157,166,174,175]. The motivation for such terms in the
case of the spreading of viruses into an uninfected population is that the time which elapses between
the moment a cell gets infected and the moment a cell gets infected and the moment a cell dies and
the virus begins to spread is an inherent part of the dynamics which cannot be neglected. Within
the context of an F-KPP type equation, these eDects would seem to lead naturally to equations with
a memory kernel, like 89
t
2
9t u = 9 x u +
dt K(t − t )u(t ) − un
(171)
or equations with a 3nite delay time $ which are obtained when K(t − t ) = (t − t − $). However,
in practice most work has concentrated on second order equations of the type [165,174,175]
$92t u + 9t u = 92x u + u − un
89
(172)
In cases in which the growth term is most naturally modeled by delay kernel, it would of course make more sense
to also take a delay-type term for the nonlinear term describing saturation [295,296]. Of course, as long as the fronts
remained pulled, this does not aDect our conclusions.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
which after appropriate rescalings are obtained by assuming that the time delay is suRciently short
that the memory kernel K can be expanded for short times as
t
dt K(t − t )u(t ) = $0 u(t) + $1 9t u(t) + $2 92t u(t) · · · ;
(173)
∞
1
d$ $ m K($) :
(174)
$m =
m! 0
Of course, the transition from a parabolic (3rst order in time) partial diDerential equation to a
hyperbolic partial diDerential equation like (172) poses new mathematical challenges concerning
existence, uniqueness and convergence [165,174,175]. However, from our pragmatic more “applied”
point of view, we want to stress the following. In practice most of the cases that are considered in
the literature on delay eDects on front propagation into an unstable state, are pulled. Furthermore,
the front velocity is the most important property one needs to know in practice. In view of this, it
is important to realize there is absolutely no reason to approximate a true time delay equation like
(172) by a hyperbolic equation by expanding the delay kernel as in (173). After all, as explained
in Section 2.4, the spreading velocity v∗ and the associated parameters ∗ and D which govern the
shape and convergence of a pulled front, can straightforwardly be determined from the more general
class of models as well. An example was given at the end of Section 2.4.
3.20. Wound healing as a front propagation problem
In the previous sections we already mentioned that front propagation into an unstable state features
in various models that have been studied in the context of biological growth and invasion problems,
as well as in the context of pulse propagation in nerves. Since there have, to our knowledge, been
few experiments in the life sciences which are directly aimed at testing some of the speci3c front
propagation predictions, we want to draw attention to a very nice recent experiment [275] on the
healing of epidermal wounds, i.e., wounds on the outer layer of the skin. In the experiments, a 4 mm
scrape wound was made; after removal of the displaced cells, the remaining cells were bathed in a
fresh culture medium and the position of the invading healing front was measured. Fig. 52 shows a
photograph of the wound while it is healing: clearly a rather well-de3ned front is seen to propagate
upward into the space made by the scrape. The data for the position of the cell front as a function
of time, extracted from such visual observations, are shown in Fig. 53 with large solid circles. As
one can see from these data, the cell front initially moves ahead fast, and then slows down (around
a time of order 40 –60 h), before it 3nally settles to a more or less constant speed.
As discussed in [275] several mechanisms of wound healing have been discussed in the literature.
Mathematically, most approaches are formulated as a system of nonlinear partial diDerential equations for cells, whose dynamics is usually taken diDusive, and whose growth responds to various
chemical signals via chemokinesis and chemotaxis, or to mechanical signals. In the most simpleminded approach, one then arrives again at a F-KPP type equation, which then incorporates the
diDusion, but lumps all the coupling to chemical and mechanical signals into a simple growth term.
For this reason, the results of the experiment on the healing of wounds were compared with numerical results for the front dynamics in the F-KPP equation, obtained by tracking the front position
at various levels of the dynamical variable u in Eq. (1) (symbols in Fig. 53); indeed, as the 3gure
illustrates, the velocity data for fronts in the F-KPP equation have some similarities with those found
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
155
Fig. 52. Photograph of a typical human cell front 10 h after wounding. The cells are mesothelial cells from the peritoneum,
the cells that form the super3cial layer of the membrane that covers the abdominal cavity and its organs. From Maini
et al. [275,276], courtesy of S. McElwain and B. MacGillavray.
experimentally: initially the velocity is relatively high, then there is a transient regime where it is
relatively low, and then it gradually approaches the asymptotic speed from below (more detailed
examples of such transient behavior and of the dependence on the level curve which is tracked, can
be found in [144]). However, the behavior in the simulations appears to be more gradual then the
rather sharp crossover seen experimentally after 40 –50 h.
It is important to keep in mind that the time-dependent velocity of pulled fronts behaves this way
quite generally—in fact, we already encountered the same behavior in the discussion of fronts in
the Taylor–Couette and Rayleigh–BIenard systems, see Fig. 22(a) of Section 3.1. As we mentioned
already there, and as will be discussed in more detail again in Section 4, for any pulled front emerging
from suRciently localized initial conditions, the asymptotic speed is approached from below. The
precise behavior at small and intermediate times does not only depend on the model, but also on
the initial conditions. In particular, when the initial conditions have large gradients (e.g., when they
show step-function like behavior), the initial front speed at almost all levels of u is initially large,
and then undershoots v∗ —see Fig. 6 of [144].
If the sharp dip in the velocity of the wound healing is a reproducible eDect, then my own guess
is that this is a sign of the importance of other eDects not included in the model, rather than a
transient behavior of an F-KPP-type model with which the data have been compared so far.
3.21. Fronts in mean 6eld approximations of growth models
In the last two decades it has become clear that many stochastic interfacial growth processes exhibit
scale invariant behavior on long time and length scales: many growth processes are characterized
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4
Distance from origin (units)
3.5
3
2.5
2
1.5
1
0.5
0
0
20
40
(a)
60
80
100
120
140
160
Time (hours)
0.04
0.035
Speed (units/hour)
0.03
0.025
0.02
0.015
0.01
0.005
0
0
(b)
20
40
60
80
100
120
140
Time (hours)
Fig. 53. (a) Crosses: typical plots from experimental data for the position of the invading cell front over diDerent substrates.
The distance units are 0:25 mm; the origin is taken as the position at 9:5 h and time is measured from that moment on.
The other symbols denote results from simulations of the F-KPP equation with a quadratic nonlinearity, in which the
front position is tracked at 1% (triangles), 10% (squares), and 50% (diamonds) of the asymptotic value. (b) Crosses:
velocity extracted from the front position data of 3gure (a). From Maini et al. [275,276], courtesy of S. McElwain and
B. MacGillavray.
by nontrivial dynamical critical exponents for the growth of the interface roughness (the root mean
square interface Kuctuations) with time and system size. There are various types of approaches
that have been used to uncover the various universality classes that govern the long-wavelength
long-time scaling, ranging from analytical mappings and solution of lattice models, to 3eld-theoretic
renormalization group calculations and extensive numerical simulations. The common denominator
of almost all of these approaches is that one starts from an appropriate coarse-grained interfacial
model. Provided such an interfacial description is appropriate, the topics that concern us in this paper
are not directly relevant for this issue, for a discussion of which we therefore refer to the reader
to the various reviews of this 3eld [22,195,239,292]. However, if the universal scaling properties
are not the main focus for a given growth problem, but if instead one wants to analyze the overall
growth shape or a growth pattern or the possibility of a morphological transition, then a mean-3eld
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
157
approximation is often an indispensable tool. In this approximation, the analysis of fronts often does
play an important role. We illustrate this general observation with an old example of a deposition
model which at the same time is a nice example of a pulled front in a very nontrivial diDerence
equation [238]. The discussion will also prepare us for an issue to which we will return later and
which is at present not completely resolved: quite often, in a mean-3eld description one arises at
pulled fronts. As we shall discuss in more detail in Sections 5 and 7, deterministic pulled fronts
do not converge to the standard type of interfacial description and as a result a standard type
mean-3eld analysis may miss some of the essential ingredients of the underlying stochastic model,
or even exhibit pathological behavior that the underlying model does not have.
In ballistic deposition, particles rain down ballistically but at random positions onto a cluster and
stick to it with a given probability as soon as they come to a site neighboring the cluster. Consider
the special case of a two-dimensional square lattice with the depositing particles coming straight
from above at discrete times (for simplicity we summarize a special case of the analysis of Krug
and Meakin [238], who consider more general dimensions and deposition under and angle). Then a
particle coming down in the ith column can stick with probability p at the 3rst site in that column
which is a nearest neighbor of the highest occupied site in that column or in the two neighboring
columns. In a mean-3eld approximation, one ignores all correlations and formulates this growth
process in terms of the probability <t (x; z) that a site at x; z is part of the deposit at time t. In this
approximation and for one-dimensional pro3les, the appropriate dynamical equation is [238]
<t+1 (z) − <t (z) = p[1 − <t (z)]{1 − [1 − <t (z − 1)][1 − <t (z)]2 }
∞
[1 − <(z )]3 :
(175)
z =z+1
The term on the left describes the change in the probability at a site at height z that is part of
the cluster; it changes when a particle is deposited at that site, and the terms on the right-hand
side model this eDect in a mean 3eld approximation. The 3rst term on the right-hand side is the
probability that that site at height z is empty, the terms between parentheses is the probability that
at least one of the neighboring sites (at height z − 1 in that column or at height z in the neighboring
columns) is occupied, and the product term is the probability that none of the sites in the column
or its two neighbors is occupied.
Eq. (175) is a diDerence equation in both the discrete space and time variables; its form is unusual, in that the change at height z depends on all the probabilities at higher sites. 90 A rather
complete study of the general form of this equation was given by Krug and Meakin [238]; their
numerical solutions showed that at long times, the dynamics leads to front type solutions. Moreover
the asymptotic velocity of these fronts turned out to be v∗ : empirically, the fronts are found to be
pulled. 91 Indeed, although as we discussed in Section 2.4 it is now clear that the linear spreading
velocity v∗ is given by the same equations for any equation which upon Fourier–Laplace transform leads to a linear equation of the form (41), this appears to be one of the earliest examples
where the pulled velocity was calculated for a nontrivial diDerence equation, and where the power
law convergence to v∗ was also tested. Indeed, by linearizing the equation in < and substituting
90
A continuum version appropriate in the limit p → 0 can be found in [43].
Intuitively, this can be understood from the observation that the nonlinearities in the dynamical equation express the
suppression of the growth due to screening.
91
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
< ∼ e+t −z , one 3nds the dispersion relation [238]
+() = ln{1 + p[e + 2]}
∗
(176)
v∗
and
can easily be obtained by calculating the minimum of +()=. As mentioned
from which
above, Krug and Meakin also veri3ed that the front velocity converged to v∗ as 1=t, with a prefactor
which was consistent with (78) to within 15% (presumably, this small discrepancy is due to the
higher order 1=t 3=2 correction, which was not known at the time). This slow convergence in time in
this case actually entails a very slow 1=z correction of the frozen-in density pro3le!
A common feature of mean 3eld approximations of growth models that (175) also exhibits is that
the growth is nonzero for arbitrarily small particle density <. This makes the state < = 0 really a
linearly unstable state and gives rise to the existence of a 3nite linear spreading speed. In reality
however, lattice models have an intrinsic cutoD for growth: there has to be at least one particle for
a cluster to be able to grow. As we shall see in Section 7, the fact that the particle occupation
number is “quantized” has important consequences for growth fronts: they are eDectively always
pushed rather than pulled.
3.22. Error propagation in extended chaotic systems
In almost all examples we have considered so far, the unstable state into which a front propagates
is a well-de3ned state which is homogeneous in the appropriate variables. 92 An exception to this
which deserves to be mentioned is the work by Kaneko [213] and Torcini et al. [396] on fronts
propagating into extended chaotic systems. The unstable state in this chase is itself a chaotic state
which is characterized by positive Lyapunov exponents. Speci3cally, the authors consider the Coupled
Map Lattice
n
xin+1 = f([1 − ]xin + 12 [xin−1 + xi+1
]) ;
(177)
where i and n are the discrete space and time variables. Note that the terms in the argument of f
proportional to have the form of a discretized version of the diDusion term. The function f maps
the interval onto itself, as is usual in studies of maps.
In general, a Lyapunov exponent of a chaotic system measures how two in3nitesimally close
initial states grow apart. In this sense a chaotic state is an unstable state, since any small perturbation
away from it grows out in time. The notion of a front introduced for such a system is illustrated
in Fig. 54 from [396]. One considers two realizations of extended chaotic states xi0 and yi0 which
diDer on one side of the system (say i 6 0) but not on the other. The front in this case is thus an
“error propagation front” in the diDerence variable xin − yin —while each individual state is chaotic
everywhere, the “error” between them spreads more and more to the right.
In the early work of Kaneko [213], the Lyapunov exponent was studied in a frame moving
with 3xed speed. The natural speed of the front was then found to be the speed at which this
velocity-dependent Lyapunov exponent was equal to zero. In our words, this means that for the
nonlinearities studied by Kaneko, the “error fronts” were pulled and that the speed v∗ was given
in terms of the Lyapunov exponent. Here we follow the formulation of Torcini et al. [396], who
92
The unstable phase-winding state into which the fronts discussed in Section 2.11.5 propagate are homogeneous in the
amplitude a and wavenumber k.
159
x-y
x
y
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Space
Fig. 54. Two initial realizations x and y of states of the coupled map lattice considered by Torcini et al. [396], together
with the diDerence x−y. Note that while each state is chaotic, the diDerence variable vanished on the right. The propagation
of the “error” x − y into this state is then an example of a front propagating into an unstable chaotic state. Both pulled
and pushed fronts can occur, depending on the form of the function f.
also studied the transition from pulled to pushed incoherent error fronts (see also [182]). One can
indeed extend the notion of a pulled front to this case by analyzing the Lyapunov exponents O()
for in3nitesimal perturbations uin of xin of the form
uin ∼ eOn−i ;
(178)
where the evolution of the uin is governed by the linearized tangent map
n
n
])([1 − ]uin + 12 [uin−1 + ui+1
]) :
uin+1 = f ([1 − ]xin + 12 [xin−1 + xi+1
(179)
Clearly, O() is the analogue for an extended chaotic system of the dispersion relation −i!(k), and
the analogue of expression (12) for the linear spreading speed v∗ is
O(∗ )
dO() ∗
vCML =
=
:
(180)
∗
d ∗
Although the precise justi3cation of this expression has not been studied, it is intuitively clear
that the basis for it is that the Lyapunov exponent de3nes a 3nite time scale O−1 (∗ ). Thus, after
averaging over times suRciently longer than O−1 (∗ ), one obtains a linear problem characterized by
an eDective dispersion relation O(), in terms of which the asymptotic (t → ∞) spreading problem
is well-de3ned.
In the numerical studies, it was found that for various functions f (corresponding to the logistic
map, the cubic map and the tent map), the error propagation fronts were pulled (in the average
sense) while in other cases (e.g., f(x) = rx mod 1 with r ¿ 1), the fronts were pushed in that their
average asymptotic speed was larger than v∗ . Clearly, these error propagation fronts are examples
of incoherent fronts, and as we noted in Section 2.7.3 we have at present no sharp mathematical
characterization of a pushed incoherent front solution—intuitively we expect that enhancement of
the growth by the nonlinearities will tend to give rise to a pushed front and that, like a coherent
pushed front solution, it falls oD spatially faster than the pulled front. But how to identify a pushed
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front other than by the empirical observation that it moves faster than v∗ I do not know. Indeed,
Torcini and co-workers [396,77] have been able to provide a reasonable guidance into what type
of maps give pushed error fronts, building on the insight from the nonlinear diDusion equation that
pushed fronts are generally associated with growth functions f which increase faster than linear for
increasing u. Possibly, further study of the pushed to pulled transition in these fronts might help to
develop a sharp de3nition of pushed incoherent fronts. For recent extensions of such chaotic front
studies, including the eDective diDusion of such fronts, the reader is referred to [397].
We note in passing that Politi and coworkers [181,348] have argued that upon coarse-graining the
eDective equation for the dynamics of the diDerence variable di =xi −yi of the lower graph of Fig. 54
becomes a diDusion-type equation with multiplicative noise, 9t d = ∇2 d + ;d where ; is a stochastic
noise term. With the Cole–Hopf transformation d=eh this equation is equivalent to the KPZ equation
[22,195,218,239]. It is amusing to realize that the arguments of Section 7.3 indicate that therefore
the scaling properties of such Kuctuating error-propagation fronts are equivalent to those of the KPZ
equation, but in one dimension higher than one would naively expect: As explained in that section,
for pulled fronts the Kuctuations in the direction of propagation continue to contribute to the scaling
behavior, instead of being integrated out.
3.23. A clock model for the largest Lyapunov exponent of the particle trajectories in a dilute gas
One present line of research in kinetic theory touching on the foundations of statistical physics
is the study of the Lyapunov exponents of the trajectories of the constituent particles or atoms in
a dilute gas (the largest Lyapunov exponent gives the rate with which two nearby initial conditions
grow apart under the dynamics, see Section 3.22 above). A few years ago van Zon et al. [430]
were able to calculate the largest Lyapunov exponent exactly in a low density hard sphere gas
by expressing it in terms of the speed of a pulled front in a diDerential-diDerence equation. The
model is actually very closely related to a computationally very eRcient lattice model introduced
by Brunet and Derrida [64,66] in the context of the connection with phase transitions in disorder
models discussed in Section 3.25.
For calculating the Lyapunov exponents, one has to analyze how the original trajectories of a
typical hard sphere grow apart from a shadow one obtained from an in3nitesimally diDerent initial
condition. For the largest Lyapunov exponent, the following picture becomes appropriate in the dilute
gas limit. When a sphere collides with another one, the diDerence after the collision between the
original trajectory of each sphere and its shadowing trajectory is, on average, equal to the largest of
the two before the collision, multiplied by a constant factor that accounts for the enlargement due
to the collision. This implies that the largest Lyapunov exponent is, in appropriate units, given by
the front speed of a “clock model”, in which every particle carries a clock with a discrete time k
which is advanced at every collision [430]. This happens according to the following rule: when two
particles collide, they both reset their respective clock values, say k and ‘, to either k + 1 or ‘ + 1,
whichever is the largest. Thus, if we denote the number of particles with clock value k by Nk , we
obtain the following dynamical equation:
dNk
=−
dt
∞
‘=−∞;‘=k
Rk; ‘ − 2Rk; k + 2
k −1
‘=−∞
Rk −1; ‘ ;
(181)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
161
where Rk; ‘ denotes the rate by which collisions occur between particles with clock values k and
‘. In the dilute gas limit correlation eDects become negligible; then Rk; ‘ is simply proportional to
Nk N‘ =N 2 when k = ‘ and to Nk2 =(2N 2 ) when two particles with equal clock value k collide. Then,
upon writing fk = Nk =N and scaling the time appropriately, we obtain
dfk
= −fk + fk2−1 + 2fk −1 Ck −2 ;
dt
= −fk + Ck2−1 − Ck2−2 ;
(182)
where we have set
Ck =
k
f‘ :
(183)
‘=−∞
Adding the equations for f‘ for all values of ‘ 6 k then yields
dCj (t)=dt = −Cj (t) + Cj2−1 (t) :
(184)
Eq. (184) has two spatially and temporally constant solutions: Cj = 0 and Cj = 1. The 3rst one
is stable and describes the state where all clocks are set to a value larger than j; the second one is
unstable and corresponds to the case where all clocks are set to values less than j. As time proceeds,
all clock values are continuously increased, and hence in the context of this model it is natural to
consider the invasion of the unstable state Cj = 1 by the state Cj = 0—as indicated above, the front
speed then determines the largest Lyapunov exponent of the dilute hard sphere gas.
Numerical solutions have shown that the fronts propagating into the unstable state Cj = 1 are
pulled fronts [430]. This in itself is maybe not so much of a surprise, as it is easy to convince
oneself that the nonlinearity on the right-hand side tends to reduce the growth relative to that of the
terms linearized about the unstable state (very much like in the F-KPP equation with a nonlinearity
of the form f(u) = u − u2 ). However, the equation does illustrate nicely a number of general points
concerning pulled fronts:
(i) The dynamical equation (184) is a diDerence-diDerential equation. Nevertheless, as we already
pointed out so often, the same equations for v∗ and for the rate of approach to v∗ (Eq. (78)
of Section 2.9) hold. The convergence of the front speed to its asymptotic value according to
this result was checked explicitly with high precision numerical simulations in [144,146].
(ii) Although the dynamics is very diDerent from that of the F-KPP equation—note the very asymmetric dynamics: the dynamics of Cj is unaDected by that of Ck with k ¿ j—the equation is
simple enough that several of the methods that play an important role in proving the existence
and convergence of fronts for the F-KPP equation (like comparison theorems) can be extended
to this equation [146].
(iii) As we will discuss in Section 7, when a mean 3eld equation is simulated with a stochastic
equation with a 3nite number of particles N , the convergence as a function of N to the asymptotic pulled v∗ of the N → ∞ mean-3eld equation is very slow, logarithmically slow. This slow
convergence was also found when the present clock model was simulated with a 3nite number
of particles [430].
(iv) In the context of the clock model the Cj ’s are non-decreasing functions of j: Cj 6 Cj+1 for any
j. However, if we just take Eq. (184) as a general dynamical equation in which the Cj ’s are
162
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
allowed to decrease with increasing j, then we 3nd that the state Cj = 1 is also convectively
unstable to a front where the Cj decrease for increasing j! This turns out to be a retracting
pulled front, a front which moves to the right in accord with the fact that any perturbation
or coherent structure solution of (184) can only move towards increasing j. Such a retracting
front corresponds to a negative value of v∗ in our equations, 93 and has an amusing property:
whereas the speed of a front which genuinely propagates into the unstable state approaches the
positive asymptotic speed from below due to the negative 1=t correction in (78), the absolute
value of the speed of a retracting front is larger than |v∗ | for large times, since v∗ itself is
negative!
(v) It is easy to modify the equation according to the general rules of thumb of Section 2.7.5 so
that the fronts become pushed; an explicit example is discussed in [146].
3.24. Propagation of a front into an unstable ferromagnetic state
Although it may not be of great practical relevance, we brieKy mention an amusing example whose
front dynamics is governed by two pulled fronts separated by a phase slip region, the dynamical
equations for an anisotropic ferromagnet [150,151]. The width of domain walls and the wavelength
of spin-wave states in ferromagnets is often large enough that a continuum approximation is justi3ed.
For a ferromagnet with an easy-plane anisotropy the free energy in dimensionless units reads [246]
1
F=
d x [(9x )2 + sin2 (9x )2 + cos2 ] :
(185)
2
Here and denote the direction of the magnetization M in polar coordinates, and we have assumed
that the magnetization only varies in the spatial x-direction. The last term in the expression shows
that the energy of states with polar angle = =2 is lowest, so indeed the free energy density
describes a situation with an easy-plane direction.
The dynamics of a ferromagnet is governed by the so-called Landau–Lifshits equations
dM
dF
dF
= 2M ×
+ M × M ×
:
(186)
dt
dM
dM
In the high damping limit, the 3rst term which describes the torque can be neglected; in appropriate
time units the equations then become
9t = 92x + [1 − (9x )2 ] sin cos ;
(187)
9t = 92x + 2(9x )(9x ) cotg :
(188)
Note that if we could consider 9x to be 3xed, the 3rst equation would be nothing but the F-KPP
equation for , with the state = =2 stable if (9x )2 ¡ 1 and unstable if (9x )2 ¿ 1. Thus a
phase-winding solution of the form = =2; = kx is unstable for k ¿ 1. Elmer et al. [150,151]
considered front propagating into this unstable state; because of the similarity of the 3rst equation
with the F-KPP equation, it is no surprise that these fronts are pulled, but the new feature is the
93
Of course, there is only one dispersion relation for Fourier modes of the dynamical equation linearized about the
unstable state. The fronts discussed above and analyzed in [430,146] fall oD as exp(−∗ j) with ∗ = 0:768 and v∗ = 4:311.
The retracting fronts correspond to the solutions of (78) which have negative ∗ = −1:609 and negative v∗ = −0:373.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
163
coupling to the -variable. 94 This front leaves behind a state with ≈ 0 but with the phase
gradient 9x essentially unaltered. Because of the strong coupling between phase and polar angle
due to the cotangent term in the -equation, this phase-winding state is unstable. As a result, the 3rst
front is followed by a region where phase slips occurs. The state which emerges from here has no
appreciable phase gradient: The nonlinear term in the -equation then Kips sign and this makes the
≈ 0 state unstable. As a result, this region is in turn invaded by a second pulled F-KPP-like front.
The propagation speed of the phase slip region can even be calculated from a conservation-type
argument for the phase winding, as all the properties of the back side of the 3rst front and of the
leading edge of the trailing front are known [150,151].
When the dynamical equations are written in terms of a complex amplitude A = sin ei , A is
found to obey a Ginzburg–Landau type equation with an unusual nonlinearity [151]. From this
perspective, the problem has some similarity with the problem of front propagation into an Eckhausunstable phase winding solution in the real Ginzburg–Landau equation that was mentioned brieKy
in Section 2.11.5.
3.25. Relation with phase transitions in disorder models
In hardly any of the situations that we have discussed so far do the front solutions which become
relevant when the initial conditions are not suRciently localized, play a role. There is one remarkable
exception to this: it was discovered by Derrida and Spohn [122] that there are some statistical
physical disorder models which can be shown to have phase transitions by mapping their generating
function onto the F-KPP equation. The two regimes of pulled fronts emerging from localized initial
conditions and of leading edge dominated dynamics associated with not suRciently localized initial
conditions then translate into two diDerent phases of the disorder model! An amusing aspect is
also that in the 3rst regime, the 1=t power law relaxation and the associated crossover behavior of
the front solutions translates back into detailed knowledge of the scaling behavior of the statistical
model. More recently, the same type of mapping was applied to the renormalization group analysis
of disordered XY models [74]—see Section 3.26.1.
Let us sketch the essence of the argument for the case of polymers on a Cayley tree with a
random potential [122]. A Cayley tree is a graph which has branches but no loops, as sketched in
Fig. 55(a). The hierarchical structure of Cayley trees makes the statistical physical models de3ned on
them amenable to detailed analysis—phase transitions in such models are usually of mean 3eld type,
due to the absence of loops, but apart from this they often catch the essence of a transition in higher
dimensions. We now consider “polymers” on such Cayley trees—we can think of them in terms of
self-avoiding random walks which step one level down in every “time” step—in the presence of a
random potential V at the bonds. These values of the potential are uncorrelated random variables
distributed according to some distribution P(V ). The statistical problem for a given realization of
potentials is then de3ned as follows in terms of the so-called partition function
Z=
exp[ − J(V1i1 + V2i2 + · · ·)] ;
(189)
walks
94
It is amusing to note that the nonlinearity of the -equation for 3xed phase gradient 9x is a special case of the
dynamical equations for the director angle in the smectic C ∗ problem discussed in Section 3.15.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
time
164
(a)
(b)
Fig. 55. (a) A “polymer” (thick lines) or self-avoiding walk on a Cayley tree. On each bond an independent random
potential is de3ned. (b) Illustration of the statistical problem of a polymer on a Cayley tree with a random potential in
the continuum limit. After Derrida and Spohn [122].
where J is the inverse temperature. The term in the exponents is the sum of all the potentials in
level 1, level 2, etc. It is convenient to consider the vertical direction as a time coordinate t and
to take the time-continuous limit. The potentials V on a branch of length dt are then taken as
independent Gaussian variables with distribution
1
V2
:
(190)
P(V ) =
exp −
(4 dt)1=2
4 dt
Because of the branching structure of the Cayley tree, the partition function obeys a simple recursion
relation,
−JV
e
Z(t)
with probability 1 − dt ;
Z(t + dt) =
(191)
e−JV [Z1 (t) + Z(2) (t)] with probability dt ;
which expresses that in the 3rst small timespan dt, the walk could remain a single walk with
probability 1 − dt, or have split into two with probability dt (the generalization to Cayley trees
where every branch splits into n ¿ 2 is obvious). Z(1) and Z(2) are the two partition functions on
these two branches.
Because the potentials are random variables, one actually has to study the distribution of partition
functions. Instead of doing this, it is more convenient [122] to study the so-called generating function
Gt (x) which encodes the average and all moments of Z,
Gt (x) ≡ exp[ − e−Jx Z(t)] ;
(192)
where the brackets indicate an average over the disorder. From this de3nition, one 3nds immediately
that for small dt
1
V2
Gt (x + V ) + dt Gt2 (x) :
Gt+dt (x) = (1 − dt) dV
exp −
(193)
(4 dt)1=2
4 dt
That the branching process simply leads to a term proportional to G 2 relies on the fact that the
partition functions Z(1) and Z(2) in (191) are independent, because on a Cayley tree two branches
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
165
do not intersect below their common branch point. In the limit dt → 0 the above equation for G
becomes simply
9t G = 92x G + G(G − 1) ;
(194)
which is nothing but the F-KPP equation (1) with quadratic nonlinearity, if we put G = 1 − u.
The remarkable 3nding is that the dynamical equation for G does not depend on the temperature
J−1 explicitly: the temperature only enters indirectly through the initial conditions which Derrida
and Spohn considered,
G0 (x) = exp(−e−Jx ) ⇔ u e−Jx
(x → ∞) :
(195)
As we know from our discussion of fronts in the F-KPP equation, when J ¿ 1, this corresponds to
localized initial conditions, and the long-time behavior of Gt (x) is then simply given by the pulled
front solution whose speed approaches v∗ according to the asymptotic formula (78). This means
that the “long-time” asymptotic average properties of the polymers are completely independent of
temperature, and that they approach an asymptotic scaling behavior independently of temperature as
well. Furthermore, when J ¡ 1, the fronts correspond to fronts whose dynamics is “leading edge
dominated” and whose velocity is determined by the initial condition, i.e., the temperature. This is
the high-temperature regime. The low-temperature (J ¿ 1) regime can be associated with a frozen
phase: roughly speaking the fact that the front speed is independent of initial conditions translates
into the free energy sticking at a constant temperature-independent value—we refer for a detailed
discussion of this and of the translation of the various front results to those for the random polymer
problem to [122]. Below in Section 3.26.1 the connection to renormalization treatments of disorder
models is discussed.
We 3nally note that a lattice model version [64,66] of the statistical–physical disorder problem
discussed here is closely related to the clock model discussed in Section 3.23.
3.26. Other examples
We 3nally draw attention to a number of issues related to front propagation which we do not
review in detail.
3.26.1. Renormalization of disorder models via traveling waves
In the previous Section 3.25 we brieKy reviewed how the generating function of a disorder model
on a Cayley tree obeys the F-KPP equation. This connection between traveling waves and disorder
models is actually more general and powerful, as was discovered by Carpentier and Le Doussal
[74,75]. Very much like (194) is the evolution equation for the generating function in the “time-wise”
direction down the Cayley tree, they found that the renormalization group equation for the distribution
function of the local fugacity variables of defects in a disordered XY model takes the form of an
F-KPP equation, with the length scale ‘ playing the role time. Like we saw in Section 3.25, the
universal behavior of this distribution function comes from the independence of v∗ on the details of
the model. Moreover, the results for the universal rate of approach to the asymptotic pulled front
speed v∗ in the case of localized initial conditions (corresponding to temperatures below the glass
transition) translate into nontrivial predictions for the scaling at the transition. We refer to these
papers for an in-depth discussion of this.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
3.26.2. Singularities and “fronts” in cascade models for turbulence
As is well known, fully developed turbulence is characterized by an energy cascade: energy Kows
from small wave numbers to larger wave numbers, till it is dissipated in the viscous range. In a shell
model for this energy Kow [408], this energy Kow down the cascade shows similarities with front
propagation. However, the essential dynamics is more properly analyzed in terms of a similarity
analysis like that for the porous medium equation [17,184,339] than in terms of fronts. If interpreted
in terms of front dynamics, these fronts have a nonlinear diDusion equation reminiscent of the one
in the bacterial growth model (147) and are pushed.
In fact, as stressed in particular by Barenblatt [24,25], there is a formal connection between
similarity analysis and front propagation, since if one write x and t of a propagating front solution
u(x − vt) in terms of logarithmic variables, x = ln X , t = ln T , a uniformly translating front solution
u(x − vt) looks like a similarity solution ũ(X=T v ). In this formulation the propagation velocity of
the front plays the role of a similarity exponent. This reformulation illustrates that in principle a
sharp distinction between front propagation and similarity analysis [24,25] is diRcult to make. In
practice, however, it is often quite clear what the most natural type of analysis is for a given problem:
especially if the system admits a linearly unstable state with a well-de3ned dispersion relation for the
unstable branch of Fourier modes, the linear spreading analysis and the associated front formulation
are the methods of choice.
3.26.3. Other biological problems
Although the clearest biological examples of front propagation into an unstable state appear to
be the population spreading phenomena mentioned in Section 3.19 and the bacterial growth patterns
discussed in Section 3.13, it is important to realize that there are many problems in biology and
biophysics which are closely related. E.g., much work on pulse propagation in bistable systems or
excitable media traces back to the study of the propagation of pulses in nerves [229,379]. Since these
are not examples of front propagation into unstable states proper, and since excellent books in which
these problems are treated have appeared recently [222,311], we will not explore the diDerences and
similarities here.
3.26.4. Solar and stellar activity cycles
There is an extensive literature—see e.g. [28] for a brief introduction and entry into the literature—
associating the sunspot cycle to the occurrence of so-called dynamo waves. The coupled equations for
the azimuthal and radial magnetic 3elds take the form of two reaction–diDusion type equations, and an
important ingredient of the relevant dynamics is associated with the problem of front propagation into
unstable states [28,395,441]. Of course, for waves on a spherical rotating object like the sun, there are
many complicating aspects: not only is there the question whether the instability is locally convective
or absolute, but also the fact that the geometry is intrinsically 3nite and that the background of the
waves is spatially varying plays a role. These latter type of issues are related to the question of the
emergence of so-called “global modes” which we will discuss brieKy in Section 6. For a detailed
discussion of the dynamo waves themselves we refer to the literature cited above.
3.26.5. Digital search trees
After acceptance of this article, we learnt of a very new exciting line of research: traveling fronts
play an important role in computer science in digital search trees and data compression. Moreover,
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
167
such types of problems are related to directed DLA on a Caley tree. See [452,453] and references
therein.
4. The mechanism underlying the universal convergence towards C∗
In Section 2.9 we saw that pulled fronts converge to their asymptotic speed and shape 95 with a
universal power law behavior—e.g., for the velocity we have according to (82)
√
3 1
3
1
∗
√
+O 2 :
(196)
Re
v(t) = v − ∗ +
2 t 2(∗ )2 t 3=2
t
D
For uniformly translating fronts, for which D is real, this expression reduces to the simpler
version (78).
In the above expression v∗ ; ∗ and D are given explicitly by Eq. (12) in terms of the dispersion
relation !(k) of the linearized dynamical equation. This and other remarkable features of this general
expression were discussed extensively in Section 2.9.3 but we did not justify or derive the result
itself. It is the aim of this section to review the underlying mechanism and derivation [144] in some
detail. The importance of the derivation does not lie in the technicalities themselves, but in the fact
that the pulled front picture that we have advanced can be made into a fully explicit and predictive
formalism.
4.1. Two important features of the linear problem
Several of the most important insights on which the full derivation of (196) is based [144] come
from our understanding of the fully linear problem discussed in Sections 2.1–2.4. In Section 2.1 we
saw that according to the saddle point analysis of the fully linear spreading problem a “steep” initial
condition gives rise to the following long-time expression for a generic dynamical 3eld ,
(; t) ∼ e
−∗ +ikr∗ −i(!r∗ −kr∗ v∗ )t e
−2 =(4Dt)
√
t
;
(197)
where = x − v∗ t, and where ∗ ; kr∗ ; v∗ and D are given by the linear spreading point equations
(12). The 3rst exponential factor is the asymptotic exponential fall-of with steepness ∗ in the
frame moving with velocity v∗ . The second exponential factor, together with the t −1=2 term, is the
3rst correction term to the asymptotic behavior—it arises simply from the Gaussian saddle point
integral. Of course, in this Gaussian term we also recognize the fundamental similarity solution of
the diDusion equation. Thus, if we de3ne a new 3eld through the transformation
(; t) ≡ e−
95
∗
+ikr∗ −i(!r∗ −kr∗ v∗ )t
(; t) ;
(198)
The shape convergence only holds for uniformly translating fronts and coherent pattern forming pulled fronts, of
course. Nevertheless, the incoherent dynamics of an incoherent pulled front arises from the intrinsic chaotic behavior of
the nonlinear state behind the front (as, e.g., in panels (a) of Figs. 16–19) or from the fact that the spreading point
dynamics does not match on to a fully coherent front pro3le in the nonlinear region (as, e.g., in Fig. 18(b)). In the
leading edge the dynamics of incoherent pulled fronts is still smooth and coherent, as close inspection of these 3gures
and Fig. 58 below shows quite clearly. This dynamics is governed by the extension of Eq. (210) below to pattern forming
fronts.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
then we expect that the long-time dynamics of
is governed by the equation
2
9
9
= D 2 + corrections ;
9t
9
(199)
2
since its fundamental similarity solution is the Gaussian form
= t −1=2 e− =4Dt of (197). Note
that the transformation (198), which was called the “leading edge transformation” in [144], acts
like a mathematical magnifying glass: it allows us to focus on the dynamics of the correction to
the dominant exponential behavior. Moreover, do realize that since the saddle point analysis is
quite general (see Section 2.4) the long-time dynamics of the leading edge variable is eDectively
governed in dominant order by this diDusion equation even if the underlying dynamical equation is
of higher order, a set of equations, a di>erence equation, etc. The reason that obeys a diDerential
equation even when the original dynamical equation is a diDerence equation is that for long times
becomes arbitrarily smooth in space and time.
The above line of reasoning is intuitive and based on working backward from the general expression (197). The diDusion-type form of the equation—3rst order in time and second order in
space—is actually an immediate consequence of the fact that two roots k coincide at the linear
spreading point k ∗ . After all, Eqs. (9) and (12) imply that in the neighborhood of the spreading
point we have in the co-moving frame ! − !∗ = −iD(k − k ∗ )2 + · · · ;
(200)
and upon inverse Fourier–Laplace transformation this gives (199) for long times. In fact, the full
generalization of (199) for is easy to derive [144]: by expanding about the linear spreading point
and taking an inverse Fourier–Laplace transform, we immediately obtain [144]
9
93
92
92
92
= D 2 + D3 3 + w
+ $2 2 + · · · ;
9t
9
9
9t 9
9t
(201)
where the expansion coeRcients D3 ; w; $2 , etc. can all be expressed in terms of the characteristic
equation of the branch corresponding to the relevant spreading point—see Eq. (5:64) of [144]. E.g.,
we simply have D3 = (1=3!) d 3 !=d k 3 |k ∗ .
Since we already know from the saddle point analysis that the relevant long-time dynamics of the
leading edge variable is a diDusion-type dynamics on slow spatial and temporal scales, the crucial
conclusion from these considerations is that 96
the dynamical equation of the leading edge variable
that governs the convergence to the
linear spreading behavior in the leading edge is a di>usion equation, with subdominant terms
which are determined explicitly in terms of the dispersion relation !(k) of the linear dynamical equation of the original problem. The √appropriate similarity variable for the long-time
expansion of is the similarity variable = t of the di>usion equation.
96
There is actually a surprise here: A priori, one would expect that the second and third terms of (201) would aDect
the subdominant t −3=2 correction term in the expression (196) for v(t). Actually, they do aDect the shape scaling function
but not the velocity correction—see Eq. (5:70) of [144]. I do not have a real intuitive understanding of this remarkable
3nding which comes out of the explicit calculation.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
169
Our second important observation concerning the fully linear problem is the following. One may
well wonder “how can the leading edge dynamics ever determine the dominant convergence dynamics
of the full nonlinear pro3le?” To understand this, let us return to (197) and write it as
−2 =(4Dt) 1
∗ e
−∗ − 2 ln t −2 =(4Dt)
;
(202)
|(; t)| ∼ e− √
=e
t where, as in (13), D−1 = Re D−1 . If we follow the position C of the level line (C ; t) = C, then
according to the above expression for the linear dynamics we have in dominant order
1
1
(203)
ln t + · · · ⇔ ˙C = − ∗ + · · · :
∗
2
2 t
We already drew attention to this logarithmic shift in our discussion of the front relaxation in
Section 2.9.1, but we now identify more clearly where this behavior originates from: The 1=t relaxation of the velocity of the level line corresponds to a logarithmic shift in the position, and this
∗
results from the Gaussian t −1=2 prefactor in combination with the overall e− spatial decay of the
leading edge variable. In other words, the prefactor of the 1=t velocity relaxation is essentially the
exponent of the appropriate similarity solution of the di>usion equation (199).
So why then does the logarithmic shift carry over to the fully nonlinear pro3le? The important
point to realize is simply that 97
C = −
when we track the position of a front whose width of the nonlinear region is 6nite, the
unbounded logarithmic shift imposed by the di>usive leading edge dynamics always dominates
the large time behavior over the relaxation of the front shape itself.
This observation was already illustrated in Fig. 12.
4.2. The matching analysis for uniformly translating fronts and coherent pattern forming fronts
So we know that the leading edge variable , which measures the deviation from the asymptotic
∗
∗
∗
∗ ∗
exponential pro3le e− +ikr −i(! −k v )t in the moving frame, obeys a diDusion-type equation with
corrections. To construct the solution to the fully nonlinear font relaxation problem, we have to
match the behavior in the leading edge (the region where the dynamical equation can be linearized
about the unstable state) to the nonlinear front region, the region where the nonlinearities in the
equation are important.
In order to understand the matching behavior, it is clearest to 3rst consider the case of uniformly
translating fronts or coherent pattern forming fronts: For these the arguments can be made most
precise and for these the matching analysis has been worked out in detail. The case of incoherent
pattern forming fronts will be discussed in the next section; the analysis given there will give
97
The argument is obvious for the divergent logarithmic term. Actually, since the front shape relaxation is driven by the
1=t term in the velocity—see Eq. (211) below—any contribution from the leading edge which gives a time-dependent shift
which decays slower than the 1=t intrinsic shape relaxation is universal. This is the reason that even the 1=t 3=2 term in the
velocity relaxation is universal: it corresponds to a shift proportional to 1=t 1=2 . Even incoherent fronts whose Kuctuating
width converges faster than 1=t to some average value, still exhibit the subdominant 1=t 3=2 velocity relaxation term.
170
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
a more intuitive dynamical argument why the leading edge variable approaches the behavior that
we identify with matching arguments here.
At the linear spreading point k ∗ two roots coincide—this is illustrated both by Eq. (200) and the
fact that v∗ corresponds to the minimum of the curve venv () in Fig. 3. It is a general result [13,14]
that in the presence of a double root the → ∞ asymptotic behavior of the uniformly translating
pulled front solutions ,v∗ () or the leading coherent pulled front solution ,v1∗ (), is
,v∗ () ∼ (a1 + a2 )e− ;
∗
,v1∗ () ∼ (A1 + A2 )e−
∗
+i(kr∗ −!r∗ t)
;
(204)
as we already noted before in Eqs. (59) and (75). As t → ∞, the leading edge dynamics should
approach this behavior, so in dominant order the matching condition for the leading edge variable
is
(; t → ∞) ∼ :
(205)
At this point, we can already understand the leading term of the convergence behavior of v(t) in a
very simply manner. For simplicity, we will from now on specialize to the case of uniformly translating fronts but the generalization to coherent pattern forming fronts is straightforward [147,383,388].
The similarity solution of the diDusion equation which for large times matches the above
∼ behavior is the “dipole solution”
∼
t 3=2
2
e− =(4Dt) ⇔ ∼ e−
∗
−3=2 ln t+ln −2 =(4Dt)
:
(206)
Analogously to the discussion following Eq. (202), this result implies that to order 1=t the velocity
relaxation is
3
+ ··· ;
(207)
2∗ t
which is indeed the leading order term of the full expression (196).
To go beyond the leading order term, we have to perform a full systematic matching calculation.
To do so, it is crucial to describe the nonlinear front in the right frame: as we saw in the previous
section, a 1=t relaxation of the velocity corresponds to an ever-increasing logarithmic shift in the
position of a level line, when viewed in the frame moving with the asymptotic speed v∗ . Thus, as
Fig. 12 illustrates so nicely, if we would attempt to do perturbation theory about the asymptotic
front solution ,v∗ (), the di>erence between the actual transient pro6le and this asymptotic pro6le
would increase without bound. Nevertheless, the shape of the transient pro3le is always close to
,v∗ placed at an appropriate position. This suggests to perturb about the asymptotic front solution
in a frame X which incorporates this shift,
v(t) = v∗ −
X = − X (t) = x − v∗ t − X (t) ;
(208)
where the shift X (t) has the expansion
2c3=2
c1 c3=2 c2
+ 3=2 + 2 + · · · : ⇔ X (t) = c1 ln t − 1=2 + · · · :
(209)
t
t
t
t
√
The expansion in powers of 1= t results from the fact that the similarity variable governing the
long-time dynamics of is X =t 1=2 , so powers of X generate powers of 1=t 1=2 .
Ẋ (t) =
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
171
From here on, the matching expansion is conceptually straightforward but technically nontrivial.
In the leading edge we make a large-t expansion in terms of the similarity variable X =t 1=2 ,
g1=2 (X =t 1=2 )
2
1=2
1=2
1=2
= t g−1=2 (X =t ) + g0 (X =t ) +
+ · · · e−X =(4Dt) ;
(210)
1=2
t
where the matching condition (205) requires that g−1=2 (X =t 1=2 ) ∼ X =t 1=2 . In the nonlinear front
region we expand the pro3le in the form dictated by the above observations, 98
;1 (X ) ;3=2 (X )
+
(x; t) = ,v∗ (X ) +
+ ··· :
(211)
t
t 3=2
These expansions generate hierarchies of equations which can be solved order by order; in 3rst
nontrivial order we straightforwardly recover the 1=t term along the same lines as above, and in the
next order the nontrivial t −3=2 term in the general expression (196) for v(t) is found [144].
The explicit calculation also shows that the functions ;1 and ;3=2 are proportional to the “shape
mode” ,v =v which gives the change in the shape of the pro3le under a change of the velocity v:
one 3nds
;1 (X ) ;3=2 (X )
,v (X ) ∗
+
,v () +
= ,v∗ (X ) +
(212)
Ẋ (t) + O(t −2 ) :
t
t 3=2
v v∗
Since Ẋ (t) is nothing but the deviation v(t) − v∗ , this implies that to order t −2 the pro3le shape can
be written in form (80), i.e.,
(x; t) = ,v(t) (X ) + O(t −2 ) :
(213)
t −2
the shape of the pro3le follows the shape of a uniformly translating
In other words, to order
pro3le with velocity equal to the instantaneous value v(t).
At 3rst sight, it may appear surprising that such a result could hold, since on the one hand v(t)
always approaches v∗ from below, and since on the other hand uniformly translating pro3les ,v with
v ¡ v∗ approach the asymptotic state = 0 in an oscillatory manner. There is no contradiction here,
however. The above expression is only the asymptotic expression for X 3xed, t → ∞. The limits
do not commute: For any 6xed time, the pro3le crosses over to expression (210) for large enough
X . As one might expect from
√ the diDusive nature of the -equation, this crossover region moves
to the right diDusively, as t [144].
These analytical predictions for the relaxation of the pro3le shape have been con3rmed in
great detail numerically for the F-KPP equation (1) with a cubic nonlinearity [144]. Fig. 56 shows
a plot of
(X ; t) − ,v∗ (X )
;
(214)
,v =v|v∗ (v∗ − v(t))
where the shape mode in the denominator was obtained by numerically solving the ordinary diDerential equation that it obeys. According to the analytical predictions, the ratio (214) should approach
unity for long time; the numerical results clearly con3rm this. Note that since the denominator
98
Remember that we have written here the form for the case that the asymptotic front solution is uniformly translating.
For coherent fronts the expansion is similar [147,383,388], we only have to add the upper indices and a phase relaxation
term.
172
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(φ−Φv*)/([v*-v(t)]δΦv/δv )
1.5
1.0
t=400
0.5
t=20
Φ
0.0
-25
0
ξX
∗
25
Fig. 56. Numerical con3rmation of the analytical prediction for the shape relaxation of a transient front in the F-KPP
equation (1) with cubic nonlinearity for various times ranging from t = 20 to t = 400. Plotted is the normalized deviation
from the asymptotic pro3le ,v∗ given by (214). As indicated by the dashed vertical line, the co-moving coordinate X is
taken such that (X ; t) = 1=2. The fact that ratio (214) converges to 1 con3rms that to order t −2 the shape of the pro3le
follows ,v(t) (X ) adiabatically. Note the diDusive-type crossover on the far right to the asymptotic behavior (210). Whether
it is accidental or signi3cant that all the lines roughly cross at one point near X ≈ 10 is at present not understood. From
Ebert et al. [144].
approaches zero for long times, plots like Fig. 56 are very accurate con3rmations of the analytical expressions—in some cases the predictions have been veri3ed to within six signi3cant 3gures
[144,146]!
As we already indicated, the matching calculation can be extended to the case of coherent pattern
forming fronts [147,383,388]. We actually already illustrated in Fig. 19 how the results of numerical
simulations of the Swift–Hohenberg equation (92) con3rm the predicted relaxation behavior of the
velocity and the wavenumber just behind the front. That the shape relaxation of the front also
follows the velocity adiabatically has also been veri3ed in these studies [147,383]. For coherent
pattern forming fronts an explicit calculation of the shape modes ,vn =v is quite nontrivial, even
numerically, since it in principle involves an in3nite set of coupled ordinary diDerential equations.
Therefore in these studies the quantity
∗
(X ; t) − n=0; ±1; ::: e−in/ t ,vn∗ (X )T
(215)
t −1
was studied. Here the brackets · T denote an average over one period T ≈ 2=/∗ in the co-moving
frame. According to the analytical predictions, the various plots of this ratio should all fall on top of
each-other, and the resulting curve is nothing but the shape mode. As Fig. 57 shows, this expectation
is borne out by the simulations. Given that for such a coherent pattern forming front obtaining the
front pro3le involves extensive interpolation, it is remarkable how nicely the scaling works [147,383]!
To our knowledge this is the only existing explicit demonstration of the universal shape relaxation
of a coherent pattern forming pulled front.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
173
0.015
normalized shape mode
0.01
0.005
0
-0.005
-0.01
-0.015
-40
-20
0
20
40
ξ
Fig. 57. Demonstration of the universal relaxation behavior of the shape of a front in the Swift–Hohenberg equation
(92), according to Spruijt et al. [147,383]. Plotted is ratio (215) for times 20 –160 in steps of 20 for fronts in numerical
simulations of fronts in for = 1=2. The fact that the various curves fall almost on top of each other con3rms that the
shape of the fronts follows the relaxation of the velocity adiabatically. The curve which deviates somewhat on the left
side is taken at the earliest time, t = 20. Up to an overall multiplicative constant, the resulting curve represents the shape
mode of the pulled Swift–Hohenberg front. Note that since is not small, various harmonics contribute to the expansion
of the asymptotic pro3le, and the front solution is not close to the front solution of the cubic amplitude equation that one
can derive in the limit 0 ¡ 1.
4.3. A dynamical argument that also holds for incoherent fronts
The matching arguments reviewed above were essentially based on the observation that the asymp∗
totic uniformly translating and coherent front solutions have a e− behavior, and that the long-time
dynamics in the leading edge should match this behavior. This essentially gave the requirement that
∼ in dominant order for long times. From there on the calculation is in essence a straightforward
matching calculation.
The above argument does not give much insight into how this linear gradient in
builds up
dynamically. Moreover, the argument hinges on the existence and behavior of a coherent pulled
front pro3le. However, the same relaxation holds for incoherent fronts. Let us now clarify why this
is so and how the linear gradient builds up dynamically [388].
The argument can be formulated quite generally, but let us for simplicity just consider the quintic
∗
∗
∗
∗ ∗
CGL equation (103). If we make a transformation to the leading edge variable =e −ikr +i(!r −kr v )t A
and write the equation for in the frame X de3ned in (208), we obtain
9t = D92X
+ N ( ) + O(t −1 ) ;
(216)
where the terms of O(t −1 ) are proportional to Ẋ (t) and come from the transformation to the frame
X , and where N ( ) denotes the nonlinear terms
N ( ) = (1 + ic3 )e−2
∗
X
| |2 − (1 − ic5 )e−4
∗
X
| |4
:
(217)
We normally associate a front with a solution for which the physical 3eld, the amplitude A in the
present case, saturates behind the front. Because of the exponential term introduced in the leading
edge transformation, this means that the leading edge variable should vanish exponentially ∼ exp X
as → −∞. Thus, to the right, for large positive X , |N | vanishes exponentially as e−2X due to
the explicit exponential term in front of the cubic term in (217), while to the left |N | vanishes
exponentially as eX because of the exponential vanishing of the leading edge variable . In other
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
II
III
I
time t–>
0
-40
III
II
time t–>
I
-20
0
ξX
20
40
0
-40
-20
0
20
40
ξX
Fig. 58. Results from simulations by Storm et al. [388] of the quintic CGL equation for the same parameter values and
run as in Fig. 19(a), shown from time 35 to 144. Panel (a) shows the nonlinear function |N | de3ned in (217) as a
function of X . Note that this function falls oD exponentially to both sides. (b) The evolution of the leading edge variable
| | for the same run as in panel (a). The region marked
I in the 3gure is the region where a linear gradient builds up,
√
II marks the crossover region which √
moves out as t and in region III is falling of in dominant order as a diDusive
Gaussian with width proportional to t. The solid line indicates the position of the maximum. The regions I, II and III
are also marked in panel (a). Note that the nonlinear function N essentially vanishes in these regions and that builds
up to the right of the region where N is non-vanishing. Intuitively, we can think of the nonlinearities as acting like an
“absorbing wall” for the 3eld .
words, |N | is actually nonzero only in some limited spatial range. This is illustrated in the space–time
plot of Fig. 58(a), which shows |N | for the same simulation of the quintic CGL as in Fig. 19(a).
The exponential vanishing to the right is clear from the 3gure; the time-dependent behavior of |N |
on the left reKects the fact that the front is incoherent.
The leading edge of the front is essentially the region where |N | is negligible. Hence it is the
semi-in3nite interval to the right of N , and this nonlinear term acts like a boundary condition for the
dynamics of in the leading edge! Moreover, since in leading order obeys a diDusion equation,
let us think of
as a density 3eld obeying a diDusion equation. In this analogy, the nonlinear
term N acts like a localized sink term or an absorbing wall which suppresses the density 3eld
(it must suppress it as behind the front
vanishes). Now, as is well known, when we have an
absorbing wall at long times the density 3eld will build up a linear gradient—from this analogy we
see that the role of the nonlinearities in the equation is to make the dynamical 3eld
build up a
linear gradient, ∼ X , at long times! In other words, the long-time behavior (205) is not only a
matching condition, but it also naturally emerges dynamically from the fact that the dynamics of
the leading edge variable is essentially governed by a diDusion equation with an absorbing wall.
Fig. 58(b) shows a space-time plot of | | from the simulations of the quintic CGL equation, which
fully con3rms the gradual buildup of a linear gradient to the right of the “absorbing wall” |N |, with
a crossover to a diDusive Gaussian behavior to the right of it.
In conclusion, we 3nally understand the universal nature of the power law corrections to the
asymptotic pulled front velocity: Only the fact that there is an absorbing wall matters, its internal
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
175
structure—i.e., the form of the nonlinearities in the equation, or the question whether the front is
uniformly translating, coherent or incoherent—is unimportant [144,388]!
5. Breakdown of moving boundary approximations of pulled fronts
So far, we have focused our discussion on the dynamics of fronts in one dimension. Quite often,
however, we are interested in the formation of patterns in two or three dimensions which naturally can
be thought of as consisting of domains separated by domain walls or fronts. Usually—think of viscous
3ngering or dendritic growth [57,73,220,226,247,249,251,338,353], bacterial growth patterns [39–41],
late stage coarsening [56,192], Kames [67,90,209,449], chemical waves [160,292,294], thermal plumes
[30,451] etc.—the state of the system inside each domain is asymptotically close to an intrinsically
stable homogeneous state. But we have also encountered examples of such type of pattern formation
in higher dimensions which are driven by the motion of a front into a linearly unstable state: streamer
patterns of Section 3.5 are formed when an ionization front propagates into a non-ionized gas, the
unstable normal state of a superconductor can give way to the superconducting state through the
motion of a front (see Section 3.8), the chaotic dynamics of domains in the rotating Rayleigh–
BIenard experiments discussed in Section 3.4 is in one regime driven by the propagation of fronts
into linearly unstable states, and if Kuctuation eDects are suRciently small, front propagation into an
unstable state can be relevant to the early stage of spinodal decomposition as well (see Section 3.7).
When the width of the fronts or domain walls separating the domains is much smaller than the
radius of curvature which is set by the typical scale of the pattern, then a natural way to analyze the
pattern dynamics is in terms of a moving boundary approximation or e>ective interface approximation. This approximation amounts to treating these fronts or transition zones as a mathematically
sharp interface or boundary. In other words, their width is taken to be zero on the outer scale of
the pattern, and their internal degrees of freedom are eliminated using singular perturbation theory.
We shall henceforth use the word boundary or interface to denote this zero width limit and use the
word front when we look at a scale where its internal structure can be resolved.
Sometimes, it is numerically advantageous go in the opposite direction, i.e., to translate a model
with sharp interfaces into what has become known as a phase 6eld model in which the order parameter 3eld varies continuously through the interfacial zone. Examples where this idea was exploited for
a variety of physical problems can be found in [29,71,187,219,221,232,372]; for careful discussions
of the derivation of a moving boundary problem for a variety of diDerent physical systems, we refer
to [30,67,132,160,219,306].
Is it true, as one might be tempted to think when looking at pictures like the streamer patterns
of Fig. 30, that a moving boundary approximation or eDective interface description can be derived
whenever the fronts in the system under study are much thinner than the scale of the pattern?
Or, to put it more clearly, can one generally derive a moving boundary approximation for front
propagation into unstable states in the limit that the width of the front is much less than its radius
of curvature? The answer to this question is a de3nite “no” for pulled fronts and “yes” for pushed
fronts [145].
How come? The reason for this statement is that for a moving boundary approximation to hold, we
need not only have separation of spatial scales but also separation of time scales: the main physical
idea underlying the derivation of a moving boundary approximation is that the front itself can on
large length and time scales be viewed as a well-de3ned coherent structure which responds on a “fast”
176
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
time scale—the internal front relaxation time—to the “slow” change in the local values of the outer
dynamical 3elds that characterize the domains. Both elements are missing for a pulled front. Indeed,
as we have seen in Section 2, the dynamics of a pulled one-dimensional front is essentially determined in the whole “semi-in3nite” region ahead of the front, not in the nonlinear transition region
itself. As a consequence, the velocity and shape of a pulled front relax very slowly, with a 1=t power
law, to their asymptotic velocity and shape. For a pushed front, on the other hand, both conditions
are ful3lled: in one dimension, a pushed front relaxes exponentially fast to its asymptotic velocity
and shape, and its dynamics responds essentially only to changes in the nonlinear front region only.
We will clarify the above statements below. We 3rst illustrate how in two and three dimensions the
relaxation correction and the curvature correction are of the same order for a spherically symmetric
expanding pulled front, and then trace the main steps of the derivation of a moving boundary
approximation to discuss the breakdown of singular perturbation theory for pulled fronts.
5.1. A spherically expanding front
It is well known—see e.g. the classic work of Allen and Cahn [6] or the reviews [56,162]—that
when the radius of curvature of a front which connects two stable states of an order parameter equation is much larger than the front width, the dominant correction to the front velocity is a curvature
correction. Consider now a front expanding radially in two or three dimensions. For large times, the
curvature correction to the front velocity will go as 1=R(t) ≈ 1=(vas t), where vas is the asymptotic
front velocity. For a front connecting two stable states, or more generally for a pushed front, this is
the only asymptotic correction. This is simply because such fronts reduce in this limit to an eDective
interface—the formal derivation is summarized in the next section—and because for a weakly curved
interface separating two states of a single order parameter the curvature is the dominant correction.
For a pulled front, on the other hand, the universal power law relaxation term is according to (78)
also of order 1=t, i.e., of the same order as the curvature correction in this case. This illustrates that
a weakly curved pulled front cannot simply be viewed as a weakly curved interface with velocity
given uniquely in terms of its local curvature: A moving boundary approximation does not hold.
We can illustrate these considerations by considering the F-KPP equation in three dimensions in
the case of spherical symmetry,
9t u(r; t) = 92r u(r; t) − (2=r)9r u + f(u) :
(218)
For a front at a distance R(t)1 from the origin, the second term on the right-hand side which
accounts for the curvature of the spherical front, serves as a small perturbation. When the front is
pushed, the singular perturbation methods discussed below show that indeed it gives a correction
−2=R(t) to the front. Thus, starting from a localized initial condition u(r; t = 0) (nonzero only in
a ball around the origin), the velocity of the pushed front for large times is v(t) ≈ v† − 2=R(t) ≈
v† − 2=(v† t) + · · ·. For a pulled front with f (0) = 1, on the other hand, one expects that for large
times one can add the curvature correction and the asymptotic relaxation term given in (78) with
v∗ = 2 and ∗ = 1, to get 99 v(t) ≈ v∗ − 2=R(t) − 3=(2t) ≈ v∗ − 5=2t + · · ·. The numerical results
shown in Fig. 59 are consistent with this asymptotic expression and hence con3rm the fact that a
moving boundary approximation can never hold for a pulled front—one cannot simply characterize
99
This result was 3rst pointed out to me by B. Derrida (private communication).
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
177
pulled front
2
curvature
1.975
correction
1.95
v(t)
=2–
v(t)
1.925
1/t
relaxation
1.9
contribution
1.875
5/
2–
t)=
v(
1.85
2t
1.825
1.8
0
0.02
0.04
(a)
0.06
0.08
0.1
1/t
pushed front
2
curvature
1.975
correction
1.95
v(t)
=2–
1/t
v(t)
1.925
1.9
1.875
1.85
1.825
1.8
(b)
0
0.02
0.04
0.06
0.08
0.1
1/t
Fig. 59. Illustration of the fact that while a moving boundary approximation applies to a weakly curved pushed front, an
interfacial description does not apply to weakly curved pulled fronts. In both graphs the full lines show the velocity as
a function of time of a spherically expanding front in the F-KPP equation, starting from initial conditions u(r; 0) = 1 for
r ¡ 1 and u(r; 0) = exp[ − (r − 1)2 ] for r ¿ 1. (a) In the pulled case, obtained with f = u − u3 , the long time relaxation
asymptotically approaches the result v(t) = 2 − 5=2t, illustrating that curvature and relaxation eDects are of the same
order. The upper dashed line indicates the relaxation that would result if only the curvature term would contribute, in
other words if a moving boundary approximation would hold. The slow convergence to the line v(t) = 2 − 5=2t is due
to the subdominant t −3=2 term in the velocity relaxation
√ expression (78). (b) Results for the pushed case, obtained with
f(u) = u + du3 − u5 with = 1=4 and d = (2 + 2= 3). According to the exact result discussed in the example at the
end of Section 2.7.1, the velocity of the pushed front for these parameters is v† = 2 (v∗ = 1 in this case). Note the rapid
convergence to the line v(t) = 2 − 2=R(t) = 2 − 1=t which is the result given by the curvature correction to the velocity,
in accord with the fact that a moving boundary approximation applies to a weakly curved pushed front.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
the properties of the weakly curved pulled front in terms of its asymptotic behavior and curvature
correction only.
5.2. Breakdown of singular perturbation theory for a weakly curved pulled front
Let us now trace the main steps of the derivation of solvability type expressions for a weakly
curved or weakly perturbed front, to illustrate the breakdown of singular perturbation theory for pulled
fronts. Our main goal is to discuss how the crucial diDerences in behavior of pulled and pushed
fronts is reKected in the derivation and in the behavior of solvability expressions that underly the
moving boundary approximation. Therefore, it suRces here to consider the simplest case, a weakly
curved front in the F-KPP equation with a slowly varying external parameter, rather than a more
general “phase 3eld model” of the type referred to in the beginning of Section 5. More detailed
discussions of the derivation of a moving boundary approximation can be found in virtually any of
the references cited there. The general and more precise discussion of the breakdown of singular
perturbation theory for pulled fronts is given by Ebert et al. [145].
Let us consider the F-KPP equation in two dimensions in the case that the nonlinear function f
depends on a small and spatially slowly varying external 3eld W ,
9u=9t = ∇2 u + f(u; W ) :
(219)
A moving boundary approximation consists of 3rst matching an inner expansion of the problem on
the scale where the fast variable (usually the order parameter variable in phase 3eld models) varies
rapidly to the outer problem on large scales, the scale of variation of the other variables like the
temperature etc. In the present case, when W is a slowly varying external 3eld, the outer problem
is simple for the stable states of f(u; W ): u will to lowest order follow the stable stationary states
of f(u; W ).
Let us imagine that the front is also curved on a large length scale, so that the curvature > of the
front is a small parameter. Since > and W are expected to be small and slowly varying, the velocity
and shape of the pro3le will diDer only weakly from the velocity and shape of the straight (planar)
front, and we expect them to vary only slowly in time and space. One therefore introduces a locally
curvilinear coordinate system, with - measuring distances normal to the front, and s distances along
the level lines of the front. In the adiabatic approximation we then perturb about the reference front
solution in the absence of curvature and W , by writing in the front region
u(x; y; t) = U0 (-) + U1 (-; s; T ) + · · · ;
(220)
where T denotes the slow outer time scale and where U0 is the planar front pro3le for W = 0 which
obeys the ordinary diDerential equation
d 2 U0
dU0
+ f(U0 ; W ) = 0 :
+ v0
2
dd-
(221)
with v0 the velocity of this pro3le. Since the strongest variation of the pro3le in the front region
is in the - direction, in the weakly curved -; s coordinate system the Laplacian becomes to lowest
order in the curvature
∇2 =
92
9
+ ··· :
+>
9-2
9-
(222)
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
179
Upon substituting (220) with this result into the F-KPP equation (219), expanding in U1 and W ,
and making the Ansatz that the velocity correction v1 of the pro3le follows the change due to the
curvature and potential adiabatically, we get
9f(U0 ; W ) dU0
−
W :
(223)
LU1 = −(v1 − >)
d9W
W =0
Here L is the linear operator
9f(U; 0) d
d2
+
;
+ v0
L≡
d-2
d9U U0
(224)
which results from linearizing about the pro3le about U0 in the frame moving with v0 .
If we perform a stability analysis of a planar front pro3le in one dimension, the same operator L
arises: for the pro3le U0 to be stable, its eigenvalues have to be non-positive. However, since the
F-KPP equation with W = 0 is translation invariant, L always has one eigenmode with eigenvalue
zero: dU0 =d- gives the change in the uniformly translating pro3le U0 if it is shifted over some small
distance a, U0 (- + a) = U0 (-) + a dU0 (-)=d- + O(a2 ), and hence it is a zero eigenmode of L,
dU0 (-)
L
=0 :
(225)
dSince the unknown function U1 in (223) is acted on by L, solvability theory (the “Fredholm alternative” in more mathematical terms) requires that the sum of the other terms in (223) are orthogonal
to this zero mode. In more technical terms, they have to be orthogonal to the “left zero mode” M0
of L, the zero mode of the adjoint L+
9f(U; 0) d
d2
+
+
(226)
L ≡
− v0
d-2
d9U U0
of L, which obeys L+ M0 = 0. Normally, it is nontrivial to 3nd the zero mode of the adjoint of a
non-self-adjoint operator. In this particular case, L can transformed into the self-adjoint Schr]odinger
operator with a simple transformation, a trick that has often been exploited in the stability analysis
[72,144,216,230,252,317,376]. Here it suRces to note the result of this analysis, which can be veri3ed
directly by substitution, namely that
dU0
:
(227)
M0 (-) = ev0 dThe solvability condition obtained from taking the inner product of (223) with M0 simply reads
∞
v0 - dU0 9f(U0 ;W )
|W =0
d9W
−∞ d- e
W :
(228)
v1 = −> −
2
∞
v0 - dU0
de
d−∞
This equation explicitly gives the change in the velocity of the front due to the curvature and the
variation of the parameter W . The 3rst term is the one we already alluded to in the previous section
on spherical fronts and gives the “motion by mean curvature” eDect familiar from many models for
coarsening [6,56,192]. The second one which gives the response to the changes in the parameter W
is of the form that one typically encounters in a solvability type approach.
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The above discussion has been based on the assumption that the curvature and front velocity
change on a slow time scale. Underlying this idea is the assumption that the shape and velocity of
the front relax exponentially on some time scale of order unity to the shape and velocity of the
planar front we perturb about. One can show this more explicitly [145] by a more careful analysis
in which the adiabatic assumption is not made immediately, and in which U1 is expanded explicitly
in terms of the eigenmodes of the stability operator L. The amplitude of these modes are than of
order $i =T , where $i are the relaxation times of these modes and T the time scale of the change of
v1 and >. Thus, if the spectrum of the stability operator L is “gapped”, i.e., if the relaxation times $i
associated with the stability operator are bounded, the separation-of-time-scales-condition underlying
the derivation of the above expression is ful3lled.
There are two ways in which the diDerence between pushed and pulled fronts shows up in the
context of the above analysis. First of all, for pushed fronts the results summarized in Section 2.8
do indeed show that the stability spectrum of a pushed front is gapped, and hence, pushed fronts
obey the conditions underlying the derivation of a moving boundary approximation following the
lines sketched above. For pulled fronts, however, the conditions are not ful3lled. That a separation
of scales is not possible is immediately clear from the fact that the velocity and shape of a planar
pulled front pro3le relax as 1=t to their asymptotic values; within the above analysis, this emerges
from the fact that the spectrum of the stability operator L of a pulled front is not “gapped”, i.e., the
relaxation times $i of the eigenmodes are arbitrarily large.
Secondly, while for pushed fronts the solvability integrals in expressions like (228) are 3nite and
well-de3ned, for pulled fronts they are in3nite. For the F-KPP equation discussed above, one can see
this as follows. As we discussed in Section 2.7.1, a pushed front falls oD for large - as exp[ − 2 -],
where 2 is the second smallest spatial decay rate. According to results (28) for the F-KPP equation
given in the example at the end of Section 2.2, for the F-KPP equation one has according to (60)
v0 + v02 − 4
- 1
− 2 ;
(229)
pushed front: U0 (-) ∼ a2 e
with 2 =
2
with v0 = v† ¿ 2. Hence the integrand of the term in the denominator of (228) converges as
√2
dU0 (-) 2
- 1
∼ e(v0 −22 )- = e− v0 −4- (v† = v0 ¿ 2) ;
(230)
e v0 dso that the solvability integral is 3nite. Likewise, if 9f(U0 ; W )=9W ∼ U0 as - → ∞, the integral
in the numerator of the second term in (228) converges. Thus, for a pushed front, the conditions
underlying the derivation of a moving boundary approximation are ful3lled, and the resulting expressions for the dependence of the velocity of the front on slowly varying parameters like W are
3nite, as it should.
It is easy to see that for pulled fronts, on the other hand, the integrand in the solvability integrals
diverges for - → ∞: indeed, according to Eq. (59) for the asymptotic decay of the pulled pro3le
U0 , we have
dU0 (-) 2 -1 2 (v0 −2∗ )v0 = -2 ;
(231)
pulled front: e
∼- e
dwith v∗ = v0 = 2. The divergence of the solvability integrals re=ects the fact that for pulled fronts
the dynamically important region is the semi-in6nite region ahead of the front, while for pushed
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181
fronts or fronts between two stable states the dynamically important region for the front dynamics
is the nonlinear front region itself. At the same time, the divergence signals the inapplicability of
singular perturbation theory for pulled fronts. 100
In this section, we have simply illustrated the breakdown of the singular perturbation theory
approach to deriving a moving boundary or eDective interface approximation for pulled fronts by
considering the simple cases of the F-KPP equation. But from our general result that pulled fronts
always relax with the same slow power law to their asymptotic speed, it is clear that this conclusion
holds more generally. Indeed, even the divergence of the solvability integrals for pulled fronts can
be derived quite generally [145].
5.3. So what about patterns generated by pulled fronts?
So far, we have focused our discussion on the fact that a moving boundary approximation does not
hold for a pulled front. What then are the implications? To my knowledge, there is at present no clear
answer to this question. Clearly, the fact that moving pulled fronts in higher dimensions can strictly
speaking never be viewed as moving interfaces, reKects that the important dynamics of a pulled
front takes place ahead of the front itself! This makes these fronts especially sensitive to changes in
the initial conditions or even to slight changes in the dynamics 101 (see Section 7). Although there
are some indications for this for the streamer patterns discussed in Section 3.5, this issue has to my
knowledge not been explored systematically. More fundamentally, the question is whether in most
cases making a moving boundary approximation in patterns dominated by propagating pulled fronts
is “almost right” in the sense that it captures all the gross qualitative features, or whether there are
new eDects which are fundamentally due to the pulled nature of the fronts. To put it concretely: are
there qualitative or quantitative diDerences in the chaotic domain dynamics of the rotating Rayleigh–
BIenard patterns of Section 3.4 between the regime where the fronts are pulled and when they are
pulled? Could it be that the behavior of the domain correlation function, or the behavior of the
correlation length and time shown in Fig. 29 for the amplitude model, is fundamentally diDerent
in the pulled and pushed regimes? Likewise, does the pulled nature of a discharge front show up
in any fundamental way in the shape and dynamics of a propagating streamer front, or does the
diDerence between the superconductor patterns of Figs. 33(a) and (b) reKect in a fundamental way
the fact that in the 3rst 3gure the front is pushed, while in the second one it might be pulled? To
my knowledge, these basic questions have not been studied at all.
One interesting example of a qualitative change in the front behavior, which is related to this issue,
is the work of Kessler and Levine [228], who show that the 3nite particle cutoD eDects discussed
in Section 7.1 may render a stable pulled front unstable. Likewise, the behavior of the average
occupation density of an ensemble of DLA 3ngers hinges on the fact that the fronts are pushed
rather then pulled, as one 3nds in a mean 3eld approximation—see the introduction of Section 7.
100
It has been proposed [83,333] to regularize the integrals by introducing a cutoD -c which is taken to in3nity at the
end of the calculation. Since then only the most divergent terms survive, this procedure reproduces trivially the change
in v∗ upon changing the linear term in f, but it does not cure the inapplicability of a moving boundary approximation
for pulled fronts [145].
101
Pulled fronts are therefore also very sensitive numerically to the scheme and grid size in the region ahead of the
front. Adaptive grid size methods that tend to put a dense grid in the nonlinear front region may not converge to the
right result. This was pointed out to me by U. Ebert (private communication).
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(a)
boundary
condition
(b)
"tail"
"front"
"bulk"
(c)
Fig. 60. (a, b) Examples of a control parameter (x) which varies in space, and for which the system is locally unstable
only in some 3nite interval [x− ; x+ ]. Case (b) is supplemented with a boundary condition A(x+ ) = 0. (c) Qualitative sketch
of a “nonlinear global mode” in the regime where (x) varies as sketched in (b) and in which the interval [x− ; x+ ] is
large enough that diDerent scaling regimes exist. The steep region on the right is close to a front solution in a spatially
uniform system.
6. Fronts and emergence of “global modes”
Throughout most of this paper, we have focused our analysis on systems which were in3nite
and homogeneous, so that their dynamical equations were spatially invariant. Especially in Kuid
dynamics systems in which there is an overall convective Kow, a crucial feature is often the fact
that the system is 3nite and that there is a well-de3ned inlet—think of the Taylor–Couette cells with
throughKow discussed in Section 3.11—or the fact that there is a well-de3ned obstacle behind which
the eDective parameters are slowly varying in space—think of the wake of the Kuid Kow behind a
body in the stream-wise direction, where as we discussed in Section 3.10 an instability can give rise
to a vortex street. Investigations of these issues in the 3eld of Kuid dynamics have given rise to the
concept of “global mode” [84,203,206,257].
The idea of a “global mode” is essentially the following. Suppose we consider a system whose
properties are slowly varying as a function of the spatial coordinate x (in the case of the Kow
behind the cylinder of Section 3.10 this would be coordinate in the stream-wise direction). As a
result of this, the dimensionless control parameter (x) is then a slowly varying function of x. The
dimensionless control parameter is de3ned as the parameter which marks that when the system
is spatially homogeneous, the basic state of the system exhibits an instability for ¿ 0, while it is
stable for ¡ 0. For pattern forming instabilities in homogeneous systems, the control parameter appears as the prefactor of the linear term in an amplitude description, see Eqs. (97) and (103).
When (x) is a slowly spatially varying function which, as sketched in Figs. 60(a) and (b), is only
positive in some 3nite range of coordinates x, then we can think of the system as being locally
unstable. Nevertheless, the system only turns globally unstable once the interval [x− ; x+ ] where
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183
(x) ¿ 0 is large enough: it is clearly not suRcient that the system is locally unstable ((x) ¿ 0 for
some x)—instability only sets in when a global mode goes unstable. While for the translationally
invariant system the unstable modes are Fourier modes, the modes which have been termed “global
modes” and which govern the instability of inhomogeneous systems are essentially localized to the
spatial range [x− ; x+ ] where (x) ¿ 0. The global modes that govern the linear instability problem
in spatially varying systems have been studied in particular for the cubic CGL equation (97) for
general (x) in the limit that the range over which (x) varies is much larger than the wavelength
of the pattern, so that a WKB analysis can be employed [84,203,206,257]. In the case sketched
in Fig. 60(b), which might be a reasonable model for the instability in the wake of a cylinder of
Section 3.10, the linear eigenmodes can be calculated explicitly. The nonlinear behavior of such
global modes has in recent years also been studied [99,100,342–344,358]. We will only highlight
here those aspects of global modes that connect immediately with the general theme of this paper.
6.1. A front in the presence of an overall convective term and a zero boundary condition
at a 6xed position
As a model problem to investigate the inKuence of a 3xed boundary in a system with a convective
instability, Couairon and Chomaz [99] have analyzed the F-KPP equation (or Ginzburg–Landau
equation for a real variable)
9t u(x; t) − s0 9x u = 92x u(x; t) + f(u);
f(u) = u + u3 − u5 ;
(232)
in the semi-in3nite axis (−∞; L], with the boundary condition
u(L; t) = 0 :
(233)
Note that the second term on the left-hand side models an overall convection with velocity s0 to
the left. 102 As we already discussed in the example at the end of Section 2.5, for an in3nite system
this term can simply be transformed away by going to a frame moving with velocity s0 to the
left; here, however, the boundary condition at the 3xed position x = L does not allow us to do
so. The speci3c choice for the function f(u) in (232) has the advantage that explicit calculations
can be done, since in the pushed regime ¡ 3=4 the pushed front solutions are known explicitly
(see the discussion of the “reduction of order method” at the end of Section 2.7.1), but is not needed
for the general scaling arguments below.
Let vsel be the selected front speed in the in3nite system for s0 = 0. If the advection velocity s0 is
larger than the selected front speed, s0 ¿ vsel , a front naturally retracts to the left and the system is
nonlinearly convectively unstable, as discussed in Section 2.10. We then expect that the asymptotic
state is simply u = 0; the explicit analysis [99] con3rms this. Of course, as we already saw in
Section 3.11, in this regime s0 ¿ vsel , the system remains very sensitive to perturbations. For example,
a 3nite forcing at x = L gives rise to a 3nite u front-type pro3le [99], while a small amount of noise
at x = L gives rise to incoherent front-type patterns. The phase diagram of convective systems with
noise has been worked out in particular by Proctor et al. [358].
If s0 ¡ vsel the in3nite system is nonlinearly absolutely unstable: virtually any initial perturbation
of the state u = 0 for x ¡ L will evolve into a front that propagates to the right, till the boundary
102
Like in our earlier sections, we use the notation s0 instead of U0 of [99] to denote the overall group velocity; moreover,
unlike in [99] the overall convection is to the left rather than to the right.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
condition at x = L makes itself felt. A full analysis in this regime con3rms our intuitive expectation
that the long-time asymptotic state is a stationary front-type pro3le locked near the right boundary.
When the advection speed is just slightly less than the selected front speed in the in3nite system,
for 0 ¡ vsel − s0 1, we expect that the 3nal stationary front pro3le remains very close to the pro3le
with speed s0 just below vsel for essentially all u ¿ 0, until a crossover behavior occurs for very
small u of O()1.
Let us 3rst make this more precise for the pushed regime, i.e., when vsel = v† . The line of analysis
we will present below generalizes many of the results of [99,100] for the speci3c case considered
there. As we noted in Section 2.7.1, the pushed front pro3le is precisely the one for which the
coeRcient a1 in the asymptotic expression (58) vanishes: a1 (v† ) = 0. Since a1 will generally go
through zero linearly in v, the uniformly translating front pro3le Us0 (-) with velocity s0 in the
in3nite system will to a good approximation in the leading edge be given by
Us0 (-1) ≈ a1 (−Wv)e−1 - + a2 (v† )e−2 - ;
|a1 |Wv −(1 −2 )† − 2 1+
;
= a2 (v )e
e
a2 (v† )
(234)
where a1 = da1 =dv and where
Wv ≡ vsel − s0 = v† − s0
(235)
is assumed to be positive but small. As we explained above, the stationary front solution in the
presence of the boundary condition u = 0 at x = L and the convective term −s0 9x u in the dynamical
equation (232) will be the one given by (234) on the outer scale till it crosses over to diDerent
behavior when u becomes of order . In this crossover range, clearly the overall term a2 e−2 - must
be of order , while the perturbative correction term will be comparable to the dominant one. The
latter implies that the exponential term between parentheses in the second line must be of order
unity in the crossover region. Upon eliminating - from the two expressions obtained from these
requirements, we obtain the scaling relation
2
∼ (Wv)J with J =
:
(236)
2 − 1
According to this expression the scaling exponent J depends on the properties of the pushed front
through the spatial decay rates 1 (v† ) and 2 (v† ). Since outside the crossover region we can to a
good approximation use the term proportional to e−2 - for the front pro3le, the distance between
the point, where the front reaches a 3nite value, and the point, where it is pinned by the boundary
condition u = 0, will for Wv → 0 scale as
1
J
ln ln Wv :
(237)
2
2
The above scaling behaviour (236) has also been obtained by Kessler et al. [227], and as they point
out this formula gives the scaling of the eDect of a 3nite particle cutoD on the front speed in the
pushed regime (see Section 7).
The scaling behavior of a pulled front is very diDerent. In fact, as it turns out the perturbation
theory about a uniformly translating pulled front in the case that the front is “cut oD” at a small value
of the 3eld by a boundary condition that makes the dynamical variable vanish, was also developed
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
185
by Brunet and Derrida [64] in the context of the stochastic fronts. We will discuss the rather peculiar
logarithmic scaling dependence in more detail in Section 7.1; the 3nal result for the relation between
Wv = v∗ − s0 and obtained from Eq. (246) below is
2 D∗
D∗
;
(238)
Wv ⇔ exp −
Wv
ln2 where as before in this paper D is the eDective diDusion coeRcient which is given in terms of the
linear dispersion relation by Eq. (13) and which according to (24) is related to the curvature of
∗
venv () at the minimum v∗ . Since the front pro3le rises approximately as e− (-−L) away from the
boundary in the outer region, the distance between the right boundary and a point where u is 3nite
will to dominant order diverge as |ln |=∗ , i.e., as
D
:
(239)
∗
Wv
The detailed matching analysis in [99] and in [100] is fully consistent with the above scaling relations
and also yields the prefactors for the speci3c nonlinearity f(u) used in (232). On the other hand,
our arguments show that these scaling results hold more generally for uniformly translating pulled
and pushed fronts which are pinned by a zero boundary condition. In fact, the arguments apply to
coherent pattern forming fronts too; indeed the result of [100] for pulled fronts in the quintic CGL
equation (103) are consistent with the above dominant scaling behavior as well.
6.2. Fronts in nonlinear global modes with slowly varying (x)
Just above the threshold to instability of a global mode, a weakly nonlinear description in terms
of the single most unstable mode suRces. However, as soon as the interval [x− ; x+ ] and (x) are
suRciently large, it is more natural to think of the nonlinear structures as being decomposed into
various regions, as sketched in Fig. 60(c): a front region where the amplitude rises quickly, a bulk
region where a well-developed pattern exists whose properties follow the local variation of (x) and
a tail region where the pattern amplitude decays back to zero. Most of the analysis of such type of
structures has been done in terms of a cubic or quintic CGL equation with a slowly spatially varying
. To emphasize the separation of scales, let us denote the slow scale on which is varying by X .
Since the CGL equation is of second order in its spatial partial derivatives, there are always two
roots q corresponding with a given amplitude a of a phase winding solution of the form A = aeiqx−i/t
of the homogeneous (translationally invariant) CGL equation—see Sections 2.11.5 and 2.11.6 and
in particular Eq. (99). Hence, when varies on a slow scales X much larger than q−1 , it becomes
possible to treat the two roots q± as slow variables q± (X ) whose variation can be treated by a nonlinear WKB type analysis—as is well known [105,155,193,413], the phase is the slow variable of the
translation invariant CGL equation, while the amplitude is slaved adiabatically to the phase dynamics.
Intuitively, we expect that when we consider structures of the type sketched in Fig. 60(c), the
region on the right should be close to a front-type solution of the type we have discussed in
this paper for translationally invariant systems, and that depending on the dynamical equation under investigation these front-like regions could be either pulled-like or pushed-like. How does this
emerge in the above type of WKB analysis? Very simply: if the translation invariant equation admits
pulled front solutions, the existence of a front type region is signaled by the two roots q± (X ) both
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
approaching the linear spreading point values k ∗ in some region [342–344], while if the translation
invariant equation admits pushed fronts [101] there is a region where the pro3le is dominated by
one of the roots which is locally close to the mode k2 of the underlying pushed front in accord with
the de3ning property (76) of a pushed coherent front solution.
For further discussion of these scenarios and of the detailed scaling analysis of the width of the
various regions we refer to the literature. In closing it is also useful to point out that while the idea
of a global mode has emerged mostly in the Kuid dynamics literature in recent years, the problem of
pattern selection through a slowly varying control parameter—often called a “ramp”—was already
considered in the 1980s both from a more general perspective and for various speci3c problems
[140,237,280,299,362,368]. Not surprisingly, there are strong similarities between these approaches
and those mentioned above.
7. Elements of stochastic fronts
We have so far limited the theoretical analysis of front propagation into unstable states to the case
in which the front dynamics is governed by deterministic dynamical equations. In this last section
on special topics we 3nally brieKy address some elements of the propagation of stochastic fronts
into an unstable state.
We do want to stress that we will only touch on a few selected topics that tie in with this review’s
main emphasis on deterministic fronts. First of all, in reading this section, one should keep in mind
that the study of stochastic fronts is a vast 3eld in itself, to which we cannot do full justice. Our
choice will be to focus on those issues that are connected with the question of how the deterministic
limit can be approached, and on how the front velocity and front diDusion coeRcient behave in that
limit. Secondly, as is well known, the critical scaling properties of Kuctuating growing interfaces
are an active research subject in itself [22,195,239]. We will not directly touch on these interesting
issues, except for a brief discussion at the end of this section of the question whether Kuctuating
pulled fronts are in a diDerent universality class from the normal type of Kuctuating interfaces.
In Section 5.2 we have seen that while singular perturbation theory works 3ne for weakly curved
pushed fronts or for one-dimensional fronts in a slowly spatially varying external 3eld, it breaks
down for pulled fronts. We saw that this implied that while the motion of a weakly curved pushed
front can be mapped onto an eDective sharp interface model, such an interfacial description does not
apply to Kuctuating pulled fronts. These 3ndings carry over to Kuctuating fronts propagating into
an unstable state: when these are pushed, their asymptotic scaling properties are simply those of an
appropriate interface model. For pulled fronts, on the other hand, the situation is more complicated:
as we shall discuss, Kuctuating variants of fronts which in the mean 3eld limit are pulled are very
sensitive to noise or to the cutoD eDects introduced inherently in simulations with discrete particles
executing stochastic jumps. We will focus our discussion on these eDects which are special to
Kuctuating fronts propagating into a linearly unstable state. A more detailed review of several of
these issues is given by Panja [332].
It is amusing to note that on hindsight, it is fair to say that the 3rst clear evidence of the strong
3nite-particle cutoD eDects on Kuctuating fronts already surfaced in 1991 [58] in the context of a
mean-3eld analysis of diDusion limited aggregation (DLA) 3ngers. It had been found empirically that
when one grows many DLA 3ngers in a channel, the level lines of the average occupation density
of these 3ngers is very close to the shape of viscous 3ngers. When researchers made attempts to
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
187
study this issue with the aid of the mean-3eld equations for this density, they ran into the problem
that the naive theory, which was based on the existence of pulled fronts in these equations [313],
leads to divergences in two dimensions. Brener et al. [58] then showed that if the nonlinear growth
term is modi3ed, the fronts behaved properly and the theory reproduced the observed behavior. The
broader implications of this empirical trick seem not to have been realized at the time. In hindsight,
this was the 3rst sign that in a 3nite-particle model like DLA, the fact that there is no growth if
there is no particle at all makes the front intrinsically pushed and that this is crucial for the model
to be well-behaved. This will be a recurrent theme in our discussion.
In the next section, we will 3rst discuss the slow convergence to the asymptotic spreading speed
∗
v as a function of N , the average number of particles in the saturated state behind the front in a
stochastic discrete particle model. Then we brieKy review some of the 3eld-theoretic formulations
of stochastic fronts, and we close with a short discussion of the implications or our 3ndings for the
asymptotic scaling properties of Kuctuating fronts in more than one dimension.
7.1. Pulled fronts as limiting fronts in di>using particle models: strong cuto> e>ects
The most natural deterministic dynamical equation that arises in the mean 3eld limit from a
stochastic model is the F-KPP equation. After all, this equation embodies the two essential ingredients
of a lattice model with discrete particles A which make diDusive hops to neighboring sites, and which
have some reaction of the type A 2A. The latter type of reaction would after a proper scaling
lead to a nonlinearity of the type f(u) = u − u2 in the F-KPP equation (1). In line with the literature
on this subject we will in this section take the F-KPP equation as our reference deterministic front
equation, but one should keep in mind that all our conclusions hold more generally.
One can arrive at the full continuum F-KPP equation from a stochastic lattice model with discrete
particles by taking the limit in which the length ‘D which a particle diDuses before giving birth to
another particle, becomes much larger than the mean inter-particle spacing. A very nice pedagogical
discussion of this limit can be found in [304]. Note that while the convergence to a F-KPP type
equation in such limits is rather obvious, the precise way in which the front diDusion coeRcient
Dfront vanishes is less trivial—stochastic particle models generally give rise to fronts which themselves
exhibit a stochastic wandering around the average front displacement, but Dfront generally depends
in an intricate way on the total front structure (see also below). We will not review the details of
such type of scaling limits here as they essentially rely on advanced methods of probability theory,
and refer to the literature [31,55,121,127,128,223,224,304] for details.
The approach to the mean 3eld limit that will interest us here in particular is the case in which
we allow an arbitrary number of particles per lattice site, but in which N , the average number of
particles per lattice site (or correlation length) in the region behind the front, becomes large. Again,
as N increases, Kuctuation eDects are suppressed and we expect a mean 3eld front equation to
emerge from the underlying stochastic model as N → ∞. Note that in this case we keep Dpart 3xed
(and often of order unity), so we cannot expect the limit to be governed by a continuum F-KPP
equation: The mean 3eld limit will be an equation for a continuum density variable u on a discrete
lattice. Luckily, this is no problem for our discussion as the concepts of pulled and pushed fronts
are equally well de3ned for diDerence equations, as we discussed in Section 2.4.
Even though earlier simulations [59] had already given hints of a very slow convergence of the
speed with N to its asymptotic value, Brunet and Derrida [64] were the 3rst to clearly identify this
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issue and to recognize that the N → ∞ convergence to the mean 3eld limit is very diDerent depending
on whether the limiting front is pulled or pushed. For pushed fronts, the nonlinear front region where
the particle density u is 3nite determines essentially the front speed. This is also reKected in the
fact that the integrands of the solvability integrals that arise in the singular perturbation theory of
Section 5.2 converge exponentially. In this 3nite-density region the relative importance of Kuctuations
decreases rapidly with N , typically as N −1=2 . Hence the convergence to the mean 3eld limit is power
law fast if the asymptotic front is pushed. For pulled fronts, on the other hand, the situation is
diDerent: The fact that they are pulled by the growth and spreading of arbitrarily small perturbations
about the unstable state implies that they are very sensitive to small changes in the dynamics in
the region of very small particle densities. At the very tip of any such front is always a region
where there are very few particles per lattice site or correlation length, and where the mean 6eld
approximation breaks down: Pulled fronts are very sensitive to the unavoidable existence of this
tip region where the particle density is of order 1=N .
Following Brunet and Derrida [64], we can understand the eDect of this on the front speed as
follows. For F-KPP type equations, the asymptotic front pro3les are uniformly translating, i.e., of
the type u(-) = u(x − vt). 103 This is true even for pulled fronts on a lattice. As we have discussed in
Section 2.2, following Eq. (25), the uniformly translating fronts with velocity v ¿ v∗ are monotonic
and fall oD simply as e−- (in other words, kr = 0), but for v ¡ v∗ the solutions in the leading edge
are of the form
uvN ≈ Re ce−-+ikr - = c1 e−- sin(kr - + c2 ) ;
(240)
where c1 and c2 are real coeRcients.
Let us now consider the case in which as N → ∞ the front is pulled, and normalize the density
3eld u such that u ≈ 1 in the region behind the front. Let us furthermore assume that the stochastic
model we consider is such that in the very tip region of the Kuctuating front—we refer to the region
where there are only very few particles per site so that the density 3eld u is of O(1=N ) at the tip of
the front—the stochastic rules are such that the growth is signi3cantly suppressed. This is what one
typically expects intuitively, as there needs to be at least one particle in order to have any growth at
all. In the case of suppression of the growth, one expects that the asymptotic average front velocity
vN approaches v∗ from below as N → ∞, and hence that for 3nite but large N the pro3le in most
of the leading edge is given to good approximation by Eq. (240) above.
For uniformly translating pro3les, we already saw in Section 2.2 that by expanding about the
linear spreading point we can for vN ¡ v∗ express kr as a function of the velocity as
∗ (v∗ − vN )
;
(241)
kr ≈
D
where we have used Eqs. (27) and (24) to write the result in terms of the eDective diDusion
coeRcient D associated with the linear spreading point. The variation of the steepness is of higher
order in v∗ − vN and to obtain the leading behavior we can therefore take ≈ ∗ ; likewise we can
in the analysis below replace the co-moving coordinate - by = x − v∗ t.
103
Remember that as in Section 2.7, we use the co-moving coordinate - = x − vt for arbitrary velocities, while = x − v∗ t.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
189
lea
din
ge
dg
e
tip
region
relative size
of fluctuations
Fig. 61. Schematic sketch of the front pro3le for a stochastic particle model for very large N . Along the vertical axis, we
plot the logarithm of the density u as a function of the co-moving spatial coordinate plotted along the horizontal axis.
The leading edge of the front, where u1, corresponds to the behavior on the right for positive . The mean 3eld pro3le
given by (243) is plotted as a full line, while the dotted area gives a qualitative idea of the importance of Kuctuation
eDects. For very large N , the Kuctuation eDects are small and can be treated perturbatively throughout most of the leading
edge, except near the tip where c (∗ )−1 ln N . As the increase in slope of the full line in this region indicates, the
3nite particle eDects suppress the growth.
Let us position this coordinate system suitably so that the nonlinear front region is roughly
∗
near ≈ 0. As N becomes large, the overall exponential behavior e− of the density 3eld u then
implies that the crossover region where u = O(1=N ) is located at
1
ln N :
(242)
∗
Moreover, as Fig. 61 illustrates, upon increasing N the leading edge of the asymptotic pro3le increases logarithmically in size, as it spans the region from small to c de3ned above. On the
other hand, the region where Kuctuations are important is roughly con6ned to a range of 6nite
size near the tip. E.g., a distance of order 5=∗ behind the tip the typical number of particles per
lattice site or correlation length is already of order 100, so their eDect is already relatively small.
Stated more clearly, for any 3nite N the leading edge exhibits a crossover at c (∗ )−1 ln N to a
Kuctuation-dominated regime but as N → ∞ the fraction of the leading edge where the Kuctuating
front pro3le is accurately approximated by the mean 3eld expression (240) approaches 100%!
The asymptotic large-N correction to the velocity now follows directly from simple matching
arguments [64]. For positive but small, at the left side of the leading edge, we know that the
∗
pro3le should converge to the e− behavior as vN ↑ v∗ and hence kr → 0. This implies c2 = 0
and c1 1=kr , so that
c u vN 1
∗
sin(kr )e− kr
(kr → 0) :
Let us now look at the crossover region at c ; Eq. (243) immediately implies
duvN =d ∗
kr cotg(kr c ) − :
uv
N
c
(243)
(244)
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
As we stated above, near and beyond the crossover scale c the deviations from the mean 3eld
expressions are signi3cant, say of order unity. In order that the above expression in the leading edge
is consistent with this, the matching condition we have to impose is that the right-hand side of the
above expression (244) deviates more than in3nitesimally from the value 1=c − ∗ it has in the
limit kr → 0, c 3xed. This is only possible if the argument of the cotangent approaches in this
limit, 104 hence if in dominant order
∗
;
(245)
kr ln N
so that according to (241) we 3nally have
2 ∗ D
+ ··· :
(246)
vN v∗ − 2
ln N
This is the central result for the dominant large-N correction that was 3rst obtained by Brunet and
Derrida [64]. 105 As our discussion shows, this expression holds generally for Kuctuating pulled fronts
which converge to a pulled front as N → ∞ and for which the Kuctuation-dominated region is in
the large-N limit con3ned to a 3nite range near the tip. 106 Just like the power law convergence to
v∗ of a deterministic front is universal and independent of the details of the model (see Sections
2.9 and 4), so is this large-N convergence: like the expressions for the temporal convergence it
only depends on ∗ and D, quantities which are completely determined by the linearized equations
of the deterministic (mean 3eld) limit! Note also that the universality is intimately related to the
logarithmic N -dependence: the only input condition for the derivation is the fact that the region
where the strength of the Kuctuations is non-vanishing remains 6nite in the limit N → ∞. Since
ln N=a ln N in dominant order, this implies that the asymptotic behavior is independent of the
precise behavior in the Kuctuation region—it does not matter whether the cutoD is eDectively at
100=N or at 1=N —the only thing that matters is that it scales as N −1 !
The above logarithmic correction to the front velocity contrasts with the power law dependence of
the correction in the pushed regime. This was 3rst pointed out by Kessler et al. [227]; the exponent
of the power law dependence is given by Eq. (236) above.
To perform an explicit calculation, one of course needs an explicit model. In their paper [64],
Brunet and Derrida did the detailed matching analysis for the deterministic F-KPP equation (1) with
a cutoD for the nonlinear growth function f of the form sketched in Fig. 62(a), i.e., f(u; ) =
S(|u| − )u(1 − un ), where = 1=N .
This deterministic front model has the advantage that many aspects can be calculated explicitly,
and that it helps one to get intuitive insight into the behavior. For instance, since there is no
growth for |u| ¡ arbitrarily small perturbations about the state u = 0 do not grow: a Gaussian
initial condition with maximum less than = 1=N just spreads out diDusively, without growing in
amplitude. Strictly speaking, therefore, the state u = 0 in this model is not linearly unstable in the
104
This is the smallest value of kr satisfying this constraint. Higher values are associated with pro3les which have
oscillations in the leading edge itself, and therefore rejected.
105
It is amusing to note that from a technical point of view the same result emerged the same year in the work of
Chomaz and Couairon [99] on global modes from a very diDerent perspective—see Eq. (238).
106
We shall see in Section 7.3 below that when the strength of Kuctuations scales with the density 3eld itself (rather than
with the square root of it), then the Kuctuations throughout the whole leading edge contribute. The Kuctuation behavior
of such fronts is in a diDerent universality class.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
(a)
191
(b)
Fig. 62. (a) The nonlinear function f(u; ) used by Brunet and Derrida [64] in the F-KPP equation to study the eDect of
a cutoD on the growth function at = 1=N . Since 1, the nonlinear behavior of f for values of u of O(1) is not visible
on this scale. (b) The nonlinear function f(u; ; r) analyzed by Panja et al. [330] (thick line). In a small interval of u of
order the growth is enhanced. For r ¡ rc = 0:283833 the fronts are pushed in the limit → 0. Stochastic simulations
con3rm this eDect.
sense we have de3ned it. An immediate consequence of this is that for = 0 fronts in this equation
cannot be of the pulled type 107 —they are pushed! In line with the fact that the stability spectrum of
a pulled front is generally continuous, while it is discrete for a pushed front, one can calculate the
relaxation times $m of the slowest relaxation modes of the fronts in the above model to 3nd [227,328]
1
2 [(m + 1)2 − 1]
$m
ln2 N
(m ¿ 1) :
(247)
The modes associated with this slow relaxation extend throughout the whole leading edge; that they
are also relevant for Kuctuating fronts is for the same reason that for N → ∞ the dominant velocity
correction is determined by the matching of the mean 3eld front pro3le to a 3nite cross-over region
where Kuctuations are important. Even though the relaxation times diverge as N → ∞ the logarithmic divergence is so slow that for many practical values of N the relaxation time is still relatively
short. If so, convergence to the asymptotic speed vN is relatively fast [227], and for all practical
purposes the front behaves like a pushed front.
Prediction (246) for the asymptotic large-N velocity correction of Kuctuating fronts has been
corroborated 108 by simulations of a variety of stochastic lattice models [59,64–66,227,329,430]; also
3eld-theoretic arguments are in agreement with this prediction [337] for reasons we shall come to
below in the next section.
For any 3nite N , a front in a stochastic model also shows a stochastic wandering about the average
position. This diDusive wandering, which for the type of models we are interested in can be studied
by tracking, e.g., the position of the point where the particle density reaches a certain fraction of
107
Keep in mind that this is only true for this particular model. If one takes instead a modi3ed models in which f(u)[email protected]
for |u| ¡ and with @ ¡ 1, then the state u = 0 is linearly unstable, but strictly speaking the linear spreading speed is
√
then equal to 2 @. Since this is less than the spreading speed to which the model converges in the limit → 0, the
velocity correction is than still given by (246). Also from this point of view we conclude that the dynamically relevant
front is a pushed one. The details of the matching analysis and of the calculation of the relaxation spectrum are diDerent
in this case, but the 3nal conclusions are the unchanged.
108
Kessler et al. [227] have reported a discrepancy of a factor of 2 in the prefactor of the 1=ln2 N correction, but the
simulations by Panja [329] do not 3nd evidence for such a large deviation.
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
the average particle density behind the front, 109 can be characterized by a front di>usion coe?cient
Dfront . The large-N asymptotic scaling of Dfront is very diRcult to study by simulations since one
needs reliable data over several decades while Dfront rapidly becomes very small. Nevertheless, Brunet
and Derrida have been able to study the scaling in simulations of a stochastic model which is closely
related to the clock model of Section 3.23, by using a clever trick which allowed them to go up to
values of N of order 10150 . They obtained a scaling
Dfront ∼ 1=ln3 N
(N → ∞) ;
(248)
and empirically found that this asymptotic scaling originates from Kuctuations in the tip region: In a
simpli3ed version of their model in which only the very 3rst lattice site of the front shows stochastic
Kuctuations, they observed the same asymptotic behavior for Dfront as in their full model. It appears
that the behavior is associated with the Kuctuation behavior of the low-lying modes of the stability
operator of the deterministic equation, whose relaxation time $m is given in (247) above. 110
The analysis we have sketched above only identi3es the leading order 3nite-N velocity correction.
This term is remarkably universal; it appears that if one wants to go beyond this term, one is forced
to perform a detailed analysis of the Kuctuations in the tip region, and to match the behavior there to
the front pro3le in the rest of the leading edge. This is a complicated problem, because the number
densities in the tip region are small so that standard perturbation methods do not apply. Although a
3rst-principles theory is still lacking, quite a bit of insight is obtained from an approximate analysis
[329] in the tip region, which focuses on the stochastic behavior of the foremost occupied lattice
site (an idea that has also played a role in earlier stochastic lattice models for small values of
N [31,223,224,331]). This analysis shows that corrections beyond the leading velocity correction
term depend on virtually all the details of the microscopic model, including the precise mean-3eld
behavior in the region behind the front. In fact, in the tip region the lattice and 3nite-particle eDects
are so important that the pro3le is not truly a uniformly translating pro3le anymore [328].
Finally, it is good to stress once more that the analysis summarized above is based on the assumption that the Kuctuation eDects in the tip region suppress the growth. Obviously, if the growth rate
for u of order 1=N is enhanced, the very tip region can move faster than v∗ of the unperturbed case,
and this will lead to a pushed front with asymptotic velocity larger than v∗ . It easy to show this
explicitly following the lines of [64] by using the growth function shown in Fig. 62(b). The explicit
analysis shows that for r ¡ rc = 0:283833 the asymptotic front speed in the limit = 1=N → 0 is
indeed larger than v∗ . Simulations of a stochastic particle model in which growth rates for particle
occupancies of 3 or less are enhanced in the same way as suggested by the function f(u; ; r) of
109
In models in which the total number of particles remains 3xed, like in the clock model of Section 3.23 or the model
studied by Brunet and Derrida [64], one can equivalently track the center of mass of the particles. However, in models
where the particle density behind the front region is 3nite, one has to be careful
√ not to include the Kuctuations from this
region—the total mass Kuctuations in this region will grow proportionally to NL where L is the size of this region. Since
L grows linearly in time, this will look like a diDusive behavior, but it has nothing to do with the di>usive wandering of
the front position that is of interest here and in the stochastic Langevin models discussed in the next section.
110
If one follows the arguments of [365] by expanding the Kuctuation behavior of the fronts in terms of the eigenmodes
of the stability operator, one 3nds that for large N the coeRcients in this expansion obey Langevin-equations with noise
which is exponentially dominated by Kuctuations at the tip. Moreover, the noise that drives the diDerent coeRcients
is correlated. It is easy to convince oneself that the resulting expressions (and Eq. (251) below) lead to logarithmic
scaling of Dfront but whether such arguments lead to behavior (248) is at present unclear to me. It is also possible that
non-perturbative eDects like those discussed in the next section are important.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
193
Fig. 62(b) already show this enhancement of the average front speed. Note that these results clearly
illustrate that the limits do not commute: if we take the limit → 0 in the function f, we arrive at
the classic F-KPP equation with asymptotic velocity v∗ = 2, while if we consider fronts with ¿ 0
and r ¡ rc and then take the limit → 0, we get a pushed front with speed v† larger than v∗ !
Since the main conclusions of this section aDect the gross features of Kuctuating fronts in more
than one dimension, we summarize them below:
Fronts in stochastic particle models which in the mean 6eld limit N → ∞ converge to a pulled
front solution, are generically pushed fronts for any 6nite N. Their speed relaxes exponentially
in time to an asymptotic speed vN which for models in which the growth at small particle
occupancies is suppressed relative to the mean 6eld value, is given for large N by (246).
Di>erent growth rules at very small particle numbers can give pushed fronts for 6nite N with
speed v† larger than the spreading speed v∗ of the mean 6eld limit.
For a 3xed N one has to consider such Kuctuating fronts as being “pushed” as far as their long-time
large-size scaling properties is concerned (see Section 7.3 below). On the other hand, all the special
scaling properties as N → ∞ originate from the fact that in this limit one approaches a pulled front
solution. To remind us of their special character, we have therefore sometimes referred to such fronts
as Kuctuating “pulled” fronts [328]. However, as we will see below, even this name does not do
full justice to the fact that in particle models, the Kuctuation properties of the fronts are essentially
like those of pushed fronts, not like those of pulled fronts.
A remarkable consequence of the eDects discussed in this section has been demonstrated by Kessler
and Levine [228]. As we discussed in Section 3.13, there are deterministic pulled fronts which do
now show a long-wavelength instability for any value of the parameters, but which do so for suitable
diDusion ratios as soon as they are pushed. As we have just seen, 3nite particle eDects are suRcient
to induce this crossover, and as was shown in [228] in this way 3nite particle eDects can induce a
long-wavelength interfacial instability.
7.2. Related aspects of =uctuating fronts in stochastic Langevin equations
The previous discussion has already mentally prepared us for another surprise concerning Kuctuating fronts in 3eld-theoretic approaches to the study of fronts propagating into an unstable state. In
a 3eld theoretic formulation, the natural starting point to study propagating fronts in the presence of
noise [10,11,129,130,177,176,259,260,289,297,337,377,409] is to start with a stochastic version of
the F-KPP equation (1),
9t u(x; t) = 92x u(x; t) + f(u(x; t)) + g(u(x; t));(x; t) :
(249)
In this Langevin-type equation ; is a stochastic noise term which is delta-correlated in space and
time with strength [email protected],
;(x; t);(x ; t ) = [email protected](x − x )(t − t ) ;
(250)
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Fig. 63. Snapshot of a stochastic
front in a simulation by Doering et al. [129] of the stochastic F-KPP equation (249) with
f(u) = u(1 − u) and g(u) = u(1 − u) (full line). With this function g(u) there are no Kuctuations in the stable state u = 1
behind the front. The dashed line is a plot of a uniformly translating front pro3le of the deterministic F-KPP equation
(1). Note that the stochastic front solution has compact support: the dynamical variable is identically zero beyond some
point. The deterministic front solution, on the other hand, is nonzero at every position.
where the brackets denote an average over the noise. Since ; is multiplied by some function g(u)
of the Kuctuating 3eld u in (249), the noise term is “multiplicative” rather than additive. 111
The form of the function g in (249) that gives the noise strength as a function of the dynamical
variable u, depends on the physical model for the noise. When the origin of the noise lies in the
Kuctuations in the
√ number of particles per correlated volume or per lattice site, then it is customary
to take g(u) ˙ u. The rationale behind this is that quite generally the Kuctuations in the number
of stochastic√moves of n independent particles in some correlated volume
or at some lattice site
√
is of order n; after normalization this gives Kuctuations of order u in the particle density. In
simulations, it is often advantageous to take the function g(u) to vanish too in the saturated state
behind the front, so that Kuctuations in the trivial state behind the front are suppressed (compare
footnote 109).
If the noise originates from independent random Kuctuations in one of the parameters of the
function f(u), it is natural to take the noise strength g(u) ˙ u for small u (see [366] and references
therein). We shall come back to this case in √
the next section.
Let us return to the case in which g(u) ˙ u for small u. The remarkable 3nding [129,130,307]
is that these stochastic fronts have compact support! In other words, these stochastic fronts are
identically zero beyond some 3nite value of x [307]. This is illustrated in Fig. 63. Since for any
3nite time front solutions in the deterministic F-KPP equation are nonzero for all x, this implies that
111
As is well known, for a stochastic diDerential equation with multiplicative noise, one faces the question whether the
noise is interpreted in the Itô or in the Stratonovich sense. The issues we address here are unaDected by the particular
choice one makes, so we refer to the literature [12,176,177] for further discussion.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
195
this type of noise has a non-perturbative e>ect on fronts—we cannot think of the noise as being a
small stochastic perturbation on top of an underlying smooth front pro3le that obeys a mean-3eld
like deterministic equation!
√
The fact that fronts in stochastic front equations with multiplicative noise with g ˙ u for small
u have compact support has as a remarkable consequence that they show the same surprising 1=ln2 N
scaling of the correction to the front velocity in the small noise limit (@ → 0 in (250)) [337]. The
reason for this is clear from the discussion of the previous section: as long as the non-perturbative
eDects of the noise is limited to a 3nite region near the tip of the front, then as we saw the 1=ln2 N
scaling of the velocity correction results from the matching analysis of the part of the leading edge
of the front where the noise eDect becomes arbitrarily small in the limit N → ∞.
There are many intriguing and open questions regarding such stochastic fronts, but we refer to the
two recent papers by Doering et al. [129,130] for further discussion of such issues. To my knowledge,
an open question is a proper understanding of scaling (248) of the front diDusion coeRcient Dfront ,
e.g., by exploiting the fact that in some limits the behavior of such Langevin equations can be
mapped onto 3nite particle models using a duality transformation [129,130,337].
We stress once more that these considerations are in practice essentially only of importance
for fronts which are pulled in the mean-3eld limit. As we mentioned earlier, the discussion of
Section 5.2 showed that singular perturbation theory does apply to pushed fronts, and this means
that in the weak noise limit one can study the eDect of noise on the motion of pushed fronts
perturbatively. As an example, we note that for small noise strength the diDusion coeRcient Dfront
of pushed fronts is given by
∞
d- e2vas - (dU0 =d-)2 g2 (U0 )
(251)
Dfront = @ −∞
2 :
∞
vas - (dU =d-)2
de
0
−∞
Here g is the multiplicative noise strength in (249) and U0 (-) = U0 (x − vas t) is the pushed front
solution of the deterministic limit of this stochastic diDerential equation [11,297,365]. This expression
has the typical structure discussed in Section 5.2, a ratio of two solvability-type integrals with the
perturbing term in the numerator—compare, e.g., Eq. (228). For pushed fronts the integrands vanish
exponentially for large positive - and hence pushed fronts are quite insensitive to the non-perturbative
eDects in the tip region when the stochastic noise is turned on. But, as we already discussed in Section
5.2, for pulled deterministic fronts the integrals do not converge; the anomalous scaling properties
of Kuctuating pulled fronts that we will discuss in the section below are a result of this. Armero
et al. [11] have compared the above expression for Dfront with extensive numerical simulations of
pushed fronts with noise, and found that the formula works quite well, even at appreciable values
of the noise strength @.
7.3. The universality class of the scaling properties of =uctuating fronts
Now that we understand some of the important features of the behavior of stochastic fronts in
one dimension, let us close with considering the implications for the universality class of the scaling
properties of these propagating fronts in more than one dimension.
As we noted in the very beginning of this section on stochastic fronts, the scaling properties of
Kuctuating front and interfaces is usually studied in terms of interface models like the KPZ equation
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W. van Saarloos / Physics Reports 386 (2003) 29 – 222
[22,195,218,239]. Now, even though in the limit N → ∞ some stochastic fronts can become pulled,
for any 3xed N or noise strength @ the stochastic fronts we have examined all are pushed fronts.
Therefore, in view of the conclusion of Section 5.2 that pushed fronts do map onto an eDective
interface model for their long-wavelength and long-time dynamics, these models should all eDectively
map onto one of the familiar type of interface models that have been studied so extensively in the
literature [22,195,218,239]. Or, to put it diDerently:
the particle
√ lattice models or the stochastic di>erential equations with multiplicative noise and
g(u) ˙ u which in the mean 6eld limit map onto dynamical equations with pulled fronts,
are in practice in the same universality classes as the usual interface models that one would
naively write down for them.
As a historical note, it is only fair to admit that I got this issue wrong initially [402]. When we started
to wonder about the question whether Kuctuating pulled fronts might be in a diDerent universality
class, we were motivated by several observations: (i) the fact that deterministic pulled fronts do not
map onto an eDective interface model; (ii) the fact that their dynamics is driven by the dynamics
in the semi-in3nite region ahead of the front itself rather than the dynamics in the front region
itself; and (iii) the fact that there were simulations of stochastic versions of the F-KPP equation
[363] which appeared to suggest that in two and higher dimensions such fronts are in a diDerent
universality class than the KPZ equation. This made us argue [402] that stochastic lattice model
realizations of the F-KPP equation with pulled fronts would exhibit non-KPZ scaling. The insight
summarized above that both lattice
√ model realizations and Langevin-type versions with multiplicative
noise strength proportional to u make the fronts always pushed invalidates this conjecture. In fact,
soon after our proposal, Moro [303] demonstrated this and presented numerical simulation results in
support of this.
There is actually one rather special case where Kuctuating fronts deserve to be viewed as genuinely
pulled, and for these the scaling properties seem to be anomalous indeed [364,403]: if one has a
medium in which the parameters entering the nonlinear growth function f(u) are Kuctuating (see,
e.g., [366]), a natural model is to take g(u) ≈ u for small u in (249). Let us consider the case that
the growth function f(u) is such that the fronts are pulled, and that f (0) = 1. In the presence of
noise the leading edge equation for a front in d bulk dimensions then reads 112
9t u = ∇2x u + (1 + ;)u :
(252)
If the front propagates on average in the x direction, then upon transforming to the co-moving frame
= x − v∗ t = x − 2t the equation for the leading edge variable = e u, which already played an
important role in the analysis of the universal relaxation of pulled fronts in Section 4.1, reads
9t = ∇ 2 +
; :
As is well known, with a Cole–Hopf transformation
KPZ equation [22,195,218,239] for h,
˜ 2+; :
9t h = ∇2 h + (∇h)
112
(253)
= eh this equation reduces to the standard
(254)
We immediately focus on the analysis in the leading edge, where nonlinear terms in u can be ignored. In Section 4.3
we showed that the nonlinear terms play the role of a boundary condition for the leading edge dynamics.
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
197
The point to note about this simple argument is that one arrives at the KPZ equation in the dimension of the system in which the front propagates: h and the noise ; are functions of both the
coordinate perpendicular to the front and the coordinates parallel to the front. This implies that
the universality class of such genuinely pulled Kuctuating fronts is that of the KPZ equation, but in
one dimension higher than one would naively guess if one would think of the front as a moving
interface [364,403]. One implication of this is that such fronts in one dimension show sub-diDusive
behavior, an observation that is corroborated by numerical simulations [364].
We emphasize again that this anomalous scaling—KPZ-type scaling but in one dimension higher
than one would naively expect—is only expected if the eDective equation is of form (252), i.e., if
the strength g(u) of the noise is linear in the dynamical 3eld u. As we discussed above, intrinsic
Kuctuations like those that arise
√ naturally in any particle model correspond naturally to Kuctuations
of strength proportional to u and in such models one indeed never sees signs of anomalous
diDusion or scaling behavior. I believe that this is due to the following. Whatever the precise model,
the relative strength g(u)=u of intrinsic Kuctuations is stronger the smaller u, i.e., the stronger the
smaller the number of particles. Now, it is easily seen from (251) that whenever g(u)=u behaves
as u−2 with 2 ¿ 0 for small u, the integrand in the numerator is exponentially dominated by the
contributions at the tip of the front. This means that fronts with intrinsic Kuctuations behave like
objects with Kuctuations in a 6nite region and hence with normal scaling behavior. In other words,
only if Kuctuations throughout the whole leading edge contribute equally to the Kuctuating behavior
(2 = 0), as they do when they are driven by Kuctuations in an external parameter (or possibly for
error propagating fronts—see the last paragraph of Section 3.22), will one see anomalous (transient)
scaling. In realistic particle models, one won’t.
These observations illustrate that the behavior of a Kuctuating front is subtle. Because they operate
close to a bifurcation point, whether or not we should think of them as being close to being pulled
depends on the quantity under consideration. Indeed, regarding their velocity, we can consider such
fronts as being “weakly pulled” for large N in the sense that their velocity approaches v∗ for N → ∞,
but when we consider the Kuctuation properties of a front with intrinsic Kuctuations we should not
think of them as being weakly pushed. 113 Hopefully future research will clarify these subtleties and
the scaling behavior (248) of Dfront .
8. Outlook
In this article we have aimed to introduce the subject of front propagation into linearly unstable
states from a unifying perspective, that allows us to bring together essentially all important developments in this 3eld. The concept of linear spreading speed v∗ not only aids in developing an intuitive
understanding and in sharply de3ning pulled and pushed fronts, but it also lies at the basis of the
formalism that allows one to derive the universal relaxation towards the pulled front speed, using
matched asymptotic expansions. In addition, “global modes”, the breakdown of an eDective interface
formulation of pulled fronts or the universal large-N velocity corrections of stochastic fronts can
all be naturally approached from this perspective. The many examples that we have discussed show
113
This was stressed in particular to me by E. Moro.
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the ubiquity of the problem of front propagation into unstable states, and illustrate how one can
understand an enormous variety of systems with a few simple tools.
There are several open issues. Concerning deterministic fronts, the most pressing one is a sharp
de3nition, using an appropriate averaging procedure, of an incoherent pushed front. In what sense
are incoherent fronts which arise from the nonexistence of coherent pattern forming fronts diDerent
from incoherent fronts which originate from the instability of coherent fronts? Furthermore, while
our de3nition of coherent pattern forming pushed fronts is consistent with what is know for the
quintic CGL equation or extensions of the Swift–Hohenberg equation, such types of fronts remain
relatively unexplored. For pulled fronts in periodic media, we have conjectured that their asymptotic
speed can be calculated using Floquet analysis, but this proposal remains to be explored. Also our
understanding of stochastic fronts, especially when they are close to the mean 3eld limit, is still
incomplete. In addition, I have the impression that the diDerence between pulled and pushed fronts
may be underestimated in the 3elds of turbulent combustion and wave propagation in periodic media.
Finally, even though we have argued that there are hints of slow convergence to the asymptotic speed
in experiments speci3cally aimed at probing pulled fronts, only very accurate new experiments can
settle this issue.
Of a diDerent nature are the challenges posed by pushed fronts. Even though the mechanism of
pushed front propagation for uniformly translating and coherent pattern forming fronts can be considered known, making concrete predictions for a given problem requires the explicit construction of
the nonlinear pushed front solution. Even for nonlinearly translating fronts, this is a highly nontrivial
problem for any equation beyond the F-KPP equation. Moreover, I do not think there is much hope
that there will ever be a general framework that allows one to calculate pushed front solutions for
large classes of equations, as every detail counts. Guessing how one can add a term to a dynamical
equation to make the front propagation become pushed is often trivial, but any serious analysis of
pushed fronts is often virtually impossible—the fact that the pushed front solutions of the quintic
CGL equation can be obtained analytically is an exceptional miracle.
For those mathematicians who only accept rigorous proofs, almost everything in this paper can
be considered an open issue. The route we have chosen to follow here is quite diDerent from the
usual one favored in the mathematics literature. I believe the approach we have advanced here is
ready to be put on rigorous footing. If this will be done, it will undoubtedly allow one to approach
large classes of equations at one fell swoop.
Acknowledgements
I would like to take this opportunity to express my thanks to Ute Ebert for intense collaborations
and discussion on front propagation issues in the last few years. Much of the presentation in this
article is shaped by the interaction with her. In addition I want to acknowledge all the other collaborators, colleagues and friends with whom I have worked on or discussed front propagation and
related issues in the past decade: Mischa Anisimov, Chiara Baggio, Markus B]ar, David Bensimon,
Henri Berestycki, Tomas Bohr, Daniel Bonn, Haim Brezis, Christiane Caroli, Jaume Casademunt,
Hugues ChatIe, Pat Cladis, Elaine Crooks, Mike Cross, Greg Dee, Patrick De Kepper, Bernard
Derrida, Arjen Doelman, George Gilmer, Ray Goldstein, Henry Greenside, Tim Halpin-Healy, Danielle
Hilhorst, Pierre Hohenberg, Robert Ho lyst, Desz]o HorvIath, Martin Howard, Joost Hulshof, Reijer
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
199
Jochemsen, Tasso Kaper, Julien Kockelkoren, Lorenz Kramer, Joachim Krug, Philip Maini, Hiroshi
Matano, Mayan Mimura, Chaouqi Misbah, Esteban Moro, Judith M]uller, Alan Newell, Yasumasa
Nishiura, Alexander Morozov, Peter PalDy-Muhoray, Debabrata Panja, Bert Peletier, Mark Peletier,
Werner Pesch, Zoltan Racz, Andrea Rocco, Vivi Rottsch]afer, Bjorn Sandstede, Boris Shraiman,
Willem Spruijt, Kees Storm, Alessandro Torcini, Goutam Tripathy, Henk van Beijeren, Jan-Bouwe
van den Berg, Martin van Hecke, Willem van de Water, Ramses van Zon, John Weeks and Rinke
Wijngaarden. Finally, I also want to thank Kees Storm for detailed comments on an earlier version
of the manuscript, Deb Panja and Alexander Morozov for help with some of the 3gures, and Yolande
van der Deijl for her perspective.
Appendix A. Comparison of the two ways of evaluating the asymptotic linear spreading problem
There are two ways to extract the long-time asymptotics of the general equation (44),
ˆ !) = Ĝ(k; !) · Ĥ (k; !) · (k;
V t = 0) ;
(k;
(A.1)
depending on whether one 3rst evaluates the !-integral of the inverse Laplace transform, or the
inverse k-integral of the Fourier transform. The 3rst route most closely parallels our earlier discussion
of Section 2.1. Indeed, if we 3rst evaluate the !-integral—which runs along a line in the upper half
!-plane, as (42) converges for suRciently large !i , so the functions are analytic suRciently far up
in the upper half !-plane—then the poles where |Ŝ| = 0 determine the dispersion relations !" (k) of
the various branches which we label by ". The remaining spatial Fourier inversion then has a form
similar to (8), except that there is an additional sum over the branch index ", and that associated
with each branch " and wavenumber k there is an eigenvector Û of the linear dynamical matrix
(see Section 5.5 of [144] further details). From there on, the analysis is essentially similar to the
one given for a single 3eld in Section 2.1 for each branch. If it so happens that there is more than
one branch where modes are unstable, then it is possible that there is more than one linear spreading
velocity v"∗ . Since we are considering a fully linear problem, it is clear that the largest value of v"∗
is the relevant linear spreading velocity for generic initial conditions which have nonzero overlap
with the associated eigenvector Û . We refer to the largest velocity of these as the linear associated
with each of these branches spreading velocity v∗ and likewise associate k ∗ , ∗ and D with the
values at the corresponding linear spreading point. 114
The advantage of this line of analysis (3rst evaluating the !-integral by contour integration) is
that the importance of initial conditions is immediately clear. In complete analogy with (37) and
the earlier discussion, so-called steep initial conditions, for which the amplitudes of the components m (x; t = 0) decay faster than exp(−∗ x), lead to pro3les which asymptotically spread with
velocity v∗ .
The second route to extract the long time asymptotics is the one which was developed since the
late 1950s in plasma physics. We only summarize the essentials here and refer to [49,62,204,264] for
details. The main idea of the analysis is the following. For each !, there are poles of the integrand
in the complex k-plane, so when ! varies along the !-integration path in the upper half of the
114
Since the discussion of Section 2.2 implies that each velocity v"∗ corresponds to the minimum of a curve v"; env , it is
sometimes said that the selected speed is the maximum of the minimum velocities. An example where this was checked
for fronts can be found in [334].
200
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
complex !-plane which runs parallel to the real axis, the poles trace out curves in the k-plane.
Imagine now that we lower the !-integration path; then these lines of poles in the k-plane will shift.
If one of these lines of poles tends to cross the k-axis, we just deform the k-integration path, so as
to avoid the lines traced out by the poles. Now, we can continue to deform the k-integration path
so as to avoid the lines of poles, until at some point, upon lowering the !-path, two lines of poles
in the k-plane come together from opposite sides of the integration path and “pinch oD” this path
[49,62,204,264]. Where will this “pinch point” be? Not surprisingly, it is the same as the saddle
point in the previous formulation. Indeed, since close to the point !"∗ , k"∗
1 d 2 ! ∗
! − !" ≈
(k − k"∗ )2 = −iD(k − k"∗ )2 ;
(A.2)
2 d k 2 k"∗
we have
k − k"∗ ≈ ± i(! − !"∗ )=D :
(A.3)
Hence as the !-integration path is lowered towards !"∗ , two poles in the k-plane merge together to
“pinch oD” the k-contour. In this analysis, which is often referred to as the “pinch point analysis”,
one thus evaluates the k-integral 3rst and then performs the !-integration. In view of (A.3), after
performing the k-integration there is a branch cut in the !-plane. Evaluation of the !-integration
around the branch cut then gives the same expressions as those we found before.
The above discussion of the pinch point analysis gives the long-time asymptotic behavior of the
Green’s function. This implicitly means that delta-function like initial conditions are assumed; the
possibility of exponentially decaying initial conditions is usually not discussed in the literature, but
these do lead to the same conclusions as before: steep initial conditions lead to pro3les spreading
asymptotically with velocity v∗ . In fact, since the k-integral is closed 3rst in the pinch-point analysis,
poles in the complex k-plane arising from exponentially decaying initial conditions give rise to
additional terms in the !-integral which compete with the contribution from the pinch point in a
way similar to the one we discussed before in Section 2.2.
One 3nal caveat is important. While the linear spreading point k ∗ is determined by a local condition, the relevance of this point for the dynamics is subject to some (mild) conditions on the
analytic structure in the k-plane. In the 3rst formulation, where we focus on the k-integral evaluation given the dispersion relation !(k), the underlying assumption is that the k-integration path can
be deformed continuously to go through the saddle point. In other words, there should be no branch
cuts or non-analyticities to prevent us from reaching this point. Furthermore, the condition that at
the spreading point D ¿ 0 is important. For, if this condition is not ful3lled, then if we would write
∗
(; t) = e− , the equation for is governed by a negative diDusion coeRcient [see Section 4.1,
Eq. (199)]. This means that convergence to a smooth asymptotic exponential behavior is not possible
for D ¡ 0.
Likewise, there are conditions on the analytic behavior in the pinch point formulation: the “pinching oD” of the deformed k-integration path by the two lines of poles in the complex k-plane means
that these two lines have to be analytical continuations of branches which are below and above the
k-integration path when it was running along the real k-axis. It actually appears to me that the precise
status of such conditions and of the relation between the conditions in the two formulations is not
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
201
fully understood. E.g., the condition D ¿ 0 is not generally found in the pinch point literature. 115
Nevertheless, our pragmatic point of view in this paper is that once can proceed with the general
analysis laid out in this paper, keeping in mind that for any given problem, one can explicitly check
whether the conditions underlying the general derivation are satis3ed.
Appendix B. Additional observations and conjectures concerning front selection
It turns out that the exact pushed front solutions of the F-KPP equation obtained by the “reduction
of order method” discussed brieKy at the end of Section 2.7.1, do have some special properties in
the complex --plane [188]. This has led to the speculation that these properties hold more generally
[188]—if true, this would allow one to obtain the pushed front speed v† without explicitly constructing the strongly heteroclinic front pro3le itself. However, the fact that pushed front solutions are
sensitive to every change in the equations indicate that such a remarkable property is very unlikely;
detailed investigations and the construction of an explicit counterexample have indeed con3rmed that
the idea is untenable [82].
Another observation which has been proposed as a possible road to understanding the selection
problem of uniformly translating fronts is to look for front solutions which monotonically connect
the exact unstable state =0 and the stable state in a 6nite but large interval [0; L] of the co-moving
coordinate - [391,392]. It is clear that this requirement does reproduce the selected speed for uniformly translating fronts. After all, this requirement amounts to searching for front solutions which
go trough zero at L when a larger interval is considered. Now, in the uniformly translating regime
the selected front is always the only front solution which is close to a front solution ,v (-) which
will just go through zero when the velocity is lowered. Indeed, the pushed front solution ,v† (-) has
a1 =0 and so in its neighborhood are the 3rst solutions (58) which have a = 0 and which go through
zero, while in the pulled case the 3rst solutions which go through zero appear just below v = v∗ ,
because the kr = 0. Thus this approach is consistent with known results for uniformly translating
fronts, but it gives no insight into the underlying dynamical mechanism and it does not apply to
pattern forming fronts. The convergence of the velocity obtained this way to the selected one can
be analyzed using the methods discussed in Section 6.1.
Recently, there has been interest in reformulating the front propagation problem in terms of
Hamilton–Jacobi theory (see, e.g., [156]). At present such methods appear to be limited to the
obtaining the asymptotic front speed, not the convergence to it. Since these results are already
contained within those discussed here, we will not discuss it further.
We 3nally note that the renormalization group approach developed for partial diDerential equations
[184] has also been used in the context of front propagation [61,83,333,334]. Viewed as a reformulation of singular perturbation theory, the renormalization group method is very useful in reinterpreting the basis of amplitude equations and similar methods [60,184]; however, as we discuss in
115
In view of (A.3) this condition implies a statement about the local orientation of the two lines of poles in the complex
k-plane: if the !-contour grazes just over !∗ in the horizontal direction the condition D ¿ 0 implies that one of the lines
of poles always lies in the upper half Wk-plane and the other one in the lower Wk-plane. The example give in [49] of
two poles which merge but which do not form a pinch point, is actually an example of a point where D ¡ 0, but whether
this is accidental I do not know. See also Appendix M of [144] for further discussion of the diDerence between the two
methods.
202
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Section 5 singular perturbation methods do not apply when one wants to go beyond the asymptotic
speed of pulled fronts, so here its use for the front propagation problem appear to be limited. In
addition, renormalization group methods can be used to study contraction to the asymptotic speed or
front stability [61]. Whether the universal convergence towards the asymptotic speed can be analyzed
or proved with this method, is to my knowledge an open question.
Appendix C. Index
Several colleagues have asked me to supply an index for this paper. Unfortunately, the paper
format of Physics Reports does not allow for an index with references to the page numbers. We
therefore provide an index to section numbers.
absolute instability
absorbing wall
adiabatic relaxation
amplitude equation
autocalytic reaction
bacterial front
ballistic deposition
bending rigidity
Benjamin–Feir instability
biological invasion/growth model
Bloch theorem
bluD body
boundary condition
Boussinesq eDect
branch cut
branches
Brillouin zone
bunching
Cahn–Hilliard equation
cascade model
Cayley tree
chaotic domain/system
chemical front
clock model
coarsening
coherent pattern forming front
coherent structure solution
combustion
comparison argument/theorem
Complex Ginzburg–Landau equation
2.5, 2.10, 3.11, 3.12, 3.17, 6
4.3
2.9.1, 2.9.2
3.1, 3.4
3.13
3.12
3.21
3.3
2.11.5, 3.11
3.13, 3.19, 3.26.3
3.18
3.10
4.3
3.2
2.4
2.4, 2.7.7
2.4
3.6
2.7.5, 2.11.3, 3.2, 3.7
3.26.2
3.25
3.4, 3.22
3.13
3.23
2.11.3, 3.14, 5
1.1, 2.7, 2.7.2, 2.8.3, 2.9.2,
2.11.2, 2.11.5, 2.11.6, 3.1, 4.2
2.11.5, 2.11.6
3.18
2.2
2.11, 2.11.5, 3.9, 3.12
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
conservation of steepness
conserved dynamics
convective instability
convective versus absolute instability
Couette Kow
counting argument
coupled map lattice
Crank–Nicholson
crossover region
cubic CGL equation
curvature correction
cutoD eDect
debunching
delay equation
dendrite
deposition model
diDerence equation
diDusion limited aggregation
director Kuctuations
discharge
discrete set
disordered XY model
dispersion relation
eDective diDusion coeRcient D
eDective interface approximation
eDects of stability
entire function
Euler approximation
exponential solutions
Extended Fisher–Komogorov equation
error propagation
facet
ferromagnet
3nite diDerence equation
3rst order phase transition
F-KPP equation
Kame
Floquet theory
Kuctuations
Kuid dynamics
fourth order equations
203
2.3
2.11.3, 3.7
1.2, 2.5, 2.10, 2.11.5, 3, 3.6, 3.9, 3.10, 3.11,
3.12, 3.17, 6
2.5, 2.10, 3.10, 3.11
3.9, 3.11
2.7.1, 2.8.1, 2.11.5
3.22
2.4
2.3
2.11.5, 3.11
5.2
7.1
3.6
3.19
3.5, 3.17, 5
3.21
2.4, 2.8.1, 2.9.1, 3.23
3.13, 3.17
3.14
3.5, 3.17
2.11.5
3.25
2.1, 2.11
2.1
3.4, 3.5, 3.13, 5, 5.1, 7
2.7.4
2.4
2.4
2.2
2.11.1
3.22
3.6
3.24
2.4, 2.9.1
3.14
1.2, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.7.1, 2.8.2
2.9.1, 2.11.1, 3.15, 3.18, 3.19, 3.23, 3.24,
3.25, 4.1, 4.2, 5.1, 5.2, 6, 7.1, 7.2
3.18, 5
3.18
3.1, 7, 7.1, 7.3
3
2.11
204
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
free energy
front diDusion coeRcient
front propagation/selection problem
global mode
generating function
Ginzburg–Landau theory
Green’s function
group velocity
Halperin–Lubensky–Ma eDect
Hamilton–Jacobi theory
hard sphere gas
heated wire experiment
hexagonal pattern
Hopf bifurcation
homoclon solution
imperfections
incoherent pattern forming front
initial conditions
instantaneous front velocity
integration contour
integro-diDerential equation
intermediate asymptotic
invasion model
kernel
kink
KPZ universality class
Kuramoto–Sivashinsky equation
Landau–Lifshitz equation
large-time asymptotics
langevin equation
leading edge
leading edge dominated dynamics
leading edge transformation
leading edge variable
level line
Levy process
liquid crystal
linearly unstable
linear marginal stability
linear spreading point
linear spreading speed
lipid
locality
2.11.3, 3.3, 3.4, 3.8, 3.14, 3.15
7.1
1.1, 2.7
1.2, 3.26.4
3.25
3.8
2.4, 2.12
2.8.1, 3.12
3.14
Appendix B
3.23
3.12
3.2
3.11, 3.12
3.12
2.5
1.1, 2.7, 2.7.3, 2.9.3, 2.11.4, 2.11.5, 2.11.6,
3.12, 3.22, 4.3
2.3
2.9.1, 2.9.2
2.4
2.4, 2.8.1
2.8.1
3.19, 3.26.3
2.4, 3.19
2.11.1, 3.15
2.11.4, 7.3
1.1, 2.7.5, 2.11.4, 2.11.6, 3.2, 3.13
3.24
2.3, 2.4, 2.11.5
7.2
1.1, 2.7.2, 4.1
2.7.6, 3.17, 3.25
4.1
4.1
2.9.1
3.18
3.14, 3.15
2.1
1.2
2.1, 2.7.7
2.1
3.3
2.6
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
localized initial condition
locking
logarithmic shift
long wavelength instability
Lyapunov exponent
Lyapunov functional
marginal stability
matching analysis
Mathieu equation
maximum growth condition
mean 3eld growth model
membrane
memory kernel
monotonic front pro3le
motion by mean curvature
moving boundary approximation
Mullins–Sekerka instability
multiplicative noise
nematic
node conservation
noise-sustained structures
non-causal
non-conserved dynamics
non-monotonic front pro3le
non-perturbative eDect
nucleation
one-parameter family
ordinary diDerential equation
Orr–Sommerfeld equation
Painleve analysis
partition function
pattern selection
pearling
periodic media
periodic front solution
phase 3eld model
phase separation
phase slip
phase winding solution
pinching
pinch point
pipe Kow
pitch
plasma
205
2.6, 2.7.6
2.4
2.9.1, 4.1
3.13
3.22, 3.23
2.7.5, 2.11.3, 3.4, 3.14
1.2, 2.8.1, 3.3
2.9.1, 4.2, 4.3, 7.1
3.18
2.2
3.21
3.3
2.4, 3.19
2.11.1, 3.5
5.2
3.4, 3.5, 3.13, 5, 5.1, 5.2, 7
3.8, 3.13, 3.17
7.2
3.14
2.11.1, 2.11.2, 2.11.3, 2.11.5
3.11, 3.12, 3.17
2.8.1
3.7
2.11.1
7.2
3.1, 3.8
2.7.1, 2.8.2, 2.11.5
2.7.1
3.9, 3.10
2.7.1
3.25
1.2
3.3
3.18
2.7
3.17, 5, 5.2
2.11.3
2.11.6, 3.8, 3.12
2.11.5, 3.1
3.3
2.1, 2.4, Appendix A
3.9
3.15
2.4, 3.5
206
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
Poiseuille Kow
pole
polymers
population dynamics
power law initial conditions
pulled front
“pulled front”
pulled versus pushed
pulse-type solution
pushed front
quintic CGL equation
ramp
Rayleigh instability
Rayleigh–Benard instability
Rayleigh–Taylor instability
reaction–diDusion model
reduction of order methods
relaxation mode
renormalization group
retracting fronts
Reynolds number
saddle point
second order phase transition
selected wavenumber
separation of scales
sidebranches
shape mode
similarity analysis
similarity variable
singular perturbation theory
sink
slowly varying control parameter
smectic
solar cycle
solvability condition
source
spark
spherically expanding front
3.3, 3.9
2.4
3.7
3.26.3
2.2
1.1, 2.6, 2.7.1, 2.7.2, 2.7.3, 2.7.5
2.9.1, 2.9.3, 2.11.1, 2.11.2, 2.11.3, 2.11.4
2.11.6, 3.4, 3.5, 3.6, 3.7, 3.12, 3.13, 3.18
3.21, 3.22, 5, 5.2, 6
7.1, 7.2, 7.3
2.6, 2.7.1, 2.7.2, 2.8.1, 2.11.6, 2.12, 3.4, 3.5,
3.13, 3.21, 3.22, 5.2
2.11.4, 2.11.5, 3.20
1.1, 2.6, 2.7.1, 2.7.2, 2.7.3, 2.7.5, 2.11.1,
2.11.2, 2.11.4, 2.11.6, 2.12, 3.4, 3.5, 3.12,
3.13, 3.18, 3.22, 5, 5.2, 6
2.7.1, 2.7.2, 2.7.5, 2.8.3, 2.11.6, 3.8, 4.3, 6
6.2
3.3, 3.8
3.1, 3.4, 3.20, 5
3.2
3.13
2.7.1, 2.7.5, 2.11.6, Appendix B
7.1
3.20, 3.26.1, Appendix B
2.11.6, 3.23
3.9
2.1, 2.8.1, 4.1
3.14
2.11.1, 2.11.5, 3.1
5, 5.2
2.11, 3.17
4.2
3.26.2, 4.1
4
Appendix B
2.11.5
6.2
3.14, 3.15
3.26.4
2.4, 5.2
2.11.5, 3.12
3.5
5.1
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
spinodal decomposition
spots
stability
steep initial conditions
steepness
stellar activity cycle
step instability
stochastic fronts
strati3ed Kuid
strongly heteroclinic orbit
structural phase transition
structural stability
subcritical bifurcation/transition
suRciently localized perturbation
superconductor
supercritical bifurcation/transition
surface tension
surface step
Swift–Hohenberg equation
switching
Taylor–Couette cell/Kow
Taylor vortex
thermal plume
thin 3lm (equations)
throughKow
tip of a front
transients
traveling wave
tricritical point
tweed pattern
two-parameter family
turbulence
Turing instability
uniformly translating front
universality classes
universal relaxation
vco
venv
von Karman vortex street
vortex
WKB-analysis
wound healing
zero mode
207
3.7, 3.8
3.9
2.8.1, 2.9.1, 5.2
2.3, 2.6, 4.1
2.2, 2.8.1
3.26.4
3.6
7
3.2
2.7.1, 2.8.1, 2.12
3.16
2.8.3, 2.11.6
2.11.6, 3.3, 3.9, 3.12
2.1
3.8
3, 3.1, 3.11, 3.12
3.3
3.6
1.1, 2.7.2, 2.11.2, 2.11.3, 3.1, 4.2
3.15
3.1, 3.11, 3.20
3.1, 3.11
5
3.2, 3.3
2.5, 3.11
7.1
2.3, 3.16, 3.20
3.11
3.14
3.16
2.7.2, 2.7.7, 2.8.2, 2.11.5
3.9, 3.18, 3.26.2
3.13
1.1, 2.2, 2.7, 2.7.1, 2.9.1, 4.2
2.6, 3.21, 7.3
2.9, 2.9.1, 2.11.2, 3.23, 3.25, 4, 4.1, 4.2,
4.3
2.3, 2.122.12
2.2, 2.4, 2.7.6, 2.7.7, 3.21, 3.23
3.9, 3.10
3.8, 3.9, 3.10
6.2
3.20
2.8.2
208
W. van Saarloos / Physics Reports 386 (2003) 29 – 222
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