Front propagation into unstable states

TL;DR

This paper reviews the theory of front propagation into unstable states, focusing on the concepts of pulled and pushed fronts, their velocities, and the universal relaxation behaviors, supported by experimental and theoretical examples.

Contribution

It provides a unified framework for understanding front propagation, clarifies the front selection problem, and derives universal power law relaxation behaviors for pulled fronts.

Findings

Pulled fronts propagate at the linear spreading velocity v*.

The approach explains the power law relaxation of transient velocity v(t).

The paper discusses the absence of a moving boundary approximation for pulled fronts.

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 definition of pulled fronts, nonlinear fronts whose asymptotic propagation speed equals v*, and pushed fronts, nonlinear fronts whose asymptotic speed v^dagger 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 unified 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.