A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations

Abstract

We propose a formal framework based on collective coordinates to reduce infinite-dimensional stochastic partial differential equations (SPDEs) with symmetry to a set of finite-dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group. We study SPDEs arising in population dynamics with multiplicative noise and additive symmetry breaking noise. The collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour, which would otherwise only be available through costly computational experiments. We corroborate our analytical results with numerical simulations of the full SPDE.

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Cited by in corpus (6)

• Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise
• Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise
• Stability of Travelling Waves for Reaction-Diffusion Equations with Multiplicative Noise
• A collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations
• Metastability of Waves and Patterns Subject to Spatially-Extended Noise
• Stability of Travelling Waves on Exponentially Long Timescales in Stochastic Reaction-Diffusion Equations