On the transition from the Ginzburg-Landau equation to the extended Fisher-Kolmogorov equation

Abstract

The GinzburgLandau GL equation generically describes the behaviour of small perturbations of a marginally unstable basic state in systems on unbounded domains In this paper we consider the transition from this generic situation to a degenerate codimension case in which the GL approach is no longer valid Instead of studying a general underlying model problem we consider a twodimensional system of coupled reactiondiusion equations in one spatial dimension We show that near the degeneration the behaviour of small perturbations is governed by the extended FisherKolmogorov eFK equation at leading order The relation between the GLequation and the eFKequation is quite subtle but can be analysed in detail The main goal of this paper is to study this relation which we do asymptotically The asymptotic analysis is compared to numerical simulations of the full reactiondiusion system As one approaches the codimension point we observe that the stable stationary periodic patterns predicted by the GLequation evolve towards various dierent families of stable stationary but not necessarily periodic socalled multibump solutions In the literature these multibump patterns are shown to exist as solutions of the eFK equation but there is no proof of the asymptotic stability of these solutions Our results suggest that these multibump patterns can also be asymptotically stable in large classes of model problems