Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations

TL;DR

This paper proves the nonlinear asymptotic stability of spatially periodic traveling waves in reaction-diffusion systems, extending existing results to nonzero wave speeds using domain techniques, with decay at a Gaussian rate.

Contribution

It introduces a novel approach using spatial domain techniques to establish stability for nonzero wave-speed traveling waves, improving upon previous methods limited to zero wave-speed cases.

Findings

Proves nonlinear stability with Gaussian decay rate for periodic traveling waves.

Extends stability results to nonzero wave-speed solutions.

Recovers and refines known zero wave-speed stability results.

Abstract

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling-waves of systems of reaction diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider's renormalization techniques do not appear to apply