Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

TL;DR

This paper demonstrates that spectrally stable traveling waves in reaction-diffusion systems remain orbitally stable under small multiplicative noise, extending stability results to systems with unequal diffusion coefficients.

Contribution

It introduces a novel approach to analyze stochastic stability of traveling waves without requiring equal diffusion coefficients, using a stochastic phase-shift and time-transform.

Findings

Traveling waves retain stability under small noise

Stability analysis applies to systems with unequal diffusion coefficients

Method extends previous semigroup approaches to stochastic settings

Abstract

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small. By applying a stochastic phase-shift together with a time-transform, we obtain a quasi-linear SPDE that describes the fluctuations from the primary wave. We subsequently follow the semigroup approach developed by Hamster and Hupkes in 2017 to handle the nonlinear stability question. The main novel feature is that we no longer require the diffusion coefficients to be equal.