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

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 in [17] to handle the nonlinear stability question. The main novel feature is that we no longer require the diffusion coefficients to be equal.

Travelling waves for reaction–diffusion equations forced by translation invariant noise

“…Stability Our first contribution is that we establish that the wave (Φ σ , c σ ) is stable, in the sense that the perturbation V (t) remains small over time scales of O(σ −2 ). In particular, we show that the semigroup techniques developed in our earlier work [16,17] are general enough to remain applicable in the present more convoluted setting. The main effort is to verify that certain technical estimates remain valid, which is possible by the powerful theory that has been developed for cylindrical Q-Wiener processes.…”

“…Phase tracking Our work here builds on the framework developed in [16,17] to study travelling waves in stochastic reaction-diffusion equations forced by a single Brownian motion. The main idea is to use a phase-tracking approach that is based purely on technical considerations rather than ad hoc geometric intuition.…”

“…In particular, (1.15) does not fit into the framework of any previous results in this area. In fact, besides our earlier work [16,17], there do not seem to be many rigorous studies of travelling waves for multi-component SPDEs in the literature.…”

“…To this end, we assume that the correlation function q is integrable, which allows us to define a bounded linear convolution operator Q : L 2 (R) → L 2 (R) that acts as (1. 16) Assuming furthermore that the Fourier transform q is a non-negative function, one can show that Q is a non-negative symmetric operator. More concretely, we have Qv, v L 2 (R) ≥ 0 for all v ∈ L 2 (R) and it is possible to define a square-root Q 1/2 : L 2 (R) → L 2 (R).…”

“…We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in [16,17] for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed.…”