A collective coordinate framework to study solitary waves in stochastically perturbed Korteweg-de Vries equations

TL;DR

This paper introduces a collective coordinate method to simplify and analyze the behavior of solitary waves in stochastically perturbed KdV equations, providing accurate predictions and insights into their stability and coherence.

Contribution

The paper develops a finite-dimensional stochastic differential equation framework to describe solitary waves in perturbed KdV equations, enhancing understanding of their dynamics and stability.

Findings

Accurately predicts solitary wave shape and location under stochastic perturbations

Estimates the time-scale for the validity of the perturbed KdV model

Identifies mechanisms of loss of coherence such as blow-up and radiation

Abstract

Stochastically perturbed Korteweg-de Vries (KdV) equations are widely used to describe the effect of random perturbations on coherent solitary waves. We present a collective coordinate approach to describe the effect on coherent solitary waves in stochastically perturbed KdV equations. The collective coordinate approach allows one to reduce the infinite-dimensional stochastic partial differential equation (SPDE) to a finite-dimensional stochastic differential equation for the amplitude, width and location of the solitary wave. The reduction provides a remarkably good quantitative description of the shape of the solitary waves and its location. Moreover, the collective coordinate framework can be used to estimate the time-scale of validity of stochastically perturbed KdV equations for which they can be used to describe coherent solitary waves. We describe loss of coherence by blow-up as well as by radiation into linear waves. We corroborate our analytical results with numerical simulations of the full SPDE.