Stability of Travelling Waves for 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 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.

References in corpus (8)

• SPDE in Hilbert Space with Locally Monotone Coefficients
• On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System
• Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations
• A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations
• A Multiscale-Analysis of Stochastic Bistable Reaction-Diffusion Equations
• $L^2$-Stability of Traveling Wave Solutions to Nonlocal Evolution Equations
• Stability of travelling waves in stochastic bistable reaction-diffusion equations
• Opening the Maslov Box for Traveling Waves in Skew-Gradient Systems

Cited by in corpus (4)

• Nonlinear stability of pulse solutions for the discrete FitzHugh-Nagumo equation with infinite-range interactions
• A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations
• Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations
• Traveling wave dynamics for Allen-Cahn equations with strong irreversibility