Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise

TL;DR

This paper demonstrates that spectrally stable travelling waves in reaction-diffusion equations remain orbitally stable under small multiplicative noise, using a stochastic phase-shift and semigroup methods.

Contribution

It introduces a novel approach combining stochastic phase-shift and semigroup techniques to analyze stability of travelling waves under multiplicative noise.

Findings

Stable travelling waves persist with small noise amplitude.

A semilinear sPDE models fluctuations around the primary wave.

The method extends deterministic stability analysis to stochastic systems.

Abstract

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable travelling 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 semilinear sPDE that describes the fluctuations from the primary wave. We subsequently develop a semigroup approach to handle the nonlinear stability question in a fashion that is closely related to modern deterministic methods.