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

TL;DR

This paper introduces a collective coordinate framework that reduces complex stochastic partial differential equations with symmetry to manageable finite-dimensional stochastic equations, accurately capturing traveling wave dynamics.

Contribution

The paper develops a novel collective coordinate method for analyzing SPDEs, enabling precise description of traveling wave shapes and dynamics with reduced computational effort.

Findings

Accurately describes traveling wave shape and diffusion in SPDEs

Provides a computationally efficient alternative to full simulations

Validates the approach with numerical simulations

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.