Conditional Speed and Shape Corrections for Travelling Wave Solutions to Stochastically Perturbed Reaction-Diffusion Systems
Mark van den Bosch, Christian H. S. Hamster, Hermen Jan Hupkes
TL;DR
This work develops a rigorous small-noise expansion framework for stochastic perturbations of planar travelling waves in reaction-diffusion SPDEs on cylindrical domains. Central to the approach is stochastic freezing, which introduces a phase shift $\Gamma(t)$ and a perturbation $V=U( x+\Gamma(t),x_\perp,t)-\Phi_0(x)$ that admits a controlled Taylor expansion $V=\sum_{j=1}^{r-1} Y_j + Z$, with a residual $Z=O(\sigma^{r-1/2})$. The authors construct expansion terms $Y_j$ via recursive integral equations, derive convolution and smoothness bounds, and establish limiting expectations of multilinear forms and smooth functionals, yielding universal expansion coefficients $h_{\infty;i}$. Conditioning on a high-probability stability event $\mathcal{A}_{\rm stb}$, they obtain a precise Taylor expansion for functionals, including the observed wave speed $C_{obs}(\sigma,\delta)$ with corrections beginning at order $\sigma^2$, and provide an algorithmic procedure to compute these coefficients. Finally, they prove nonlinear stability of the expansion on timescales exponentially long in $1/\sigma$, using a time-transformation and mild formulation to bound the residual and higher-order terms, thereby giving rigorous stochastic corrections to both wave speed and shape with explicit error control.
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.