Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems

TL;DR

This paper develops a rigorous mathematical framework to analyze how small stochastic perturbations affect the speed and shape of travelling wave solutions in reaction-diffusion systems, providing explicit correction formulas.

Contribution

It introduces a stochastic freezing method for reaction-diffusion waves, enabling precise calculation of stochastic corrections to wave speed and shape to arbitrary order.

Findings

Derived explicit formulas for stochastic speed corrections

Established a method for high-order noise impact analysis

Provided rigorous definitions for stochastic wave modifications

Abstract

In this work we perform rigorous small noise expansions to study the impact of stochastic forcing on the behaviour of planar travelling wave solutions to reaction-diffusion equations on cylindrical domains. In particular, we use a stochastic freezing approach that allows effective limiting information to be extracted concerning the behaviour of the stochastic perturbations from the deterministic wave. As an application, this allows us to provide a rigorous definition for the stochastic corrections to the wave speed. In addition, our approach allows their size to be computed to any desired order in the noise strength, provided that sufficient smoothness is available.