Soliton dynamics for the Korteweg-de Vries equation with multiplicative homogeneous noise

TL;DR

This paper studies how solitons in the Korteweg-de Vries equation behave under small, random, space-time dependent perturbations, deriving modulation equations, exit times, and a central limit theorem for the dispersive component.

Contribution

It provides a detailed analysis of soliton dynamics under stochastic perturbations, including precise estimates and asymptotic behavior, which advances understanding of stochastic KdV equations.

Findings

Estimated exit times from soliton neighborhoods.

Derived modulation equations for soliton parameters.

Proved a central limit theorem for the dispersive part.

Abstract

We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the dynamics of the soliton of the KdV equation in the presence of this random perturbation, assuming that the amplitude of the perturbation is small. We estimate precisely the exit time of the perturbed solution from a neighborhood of the modulated soliton, and we obtain the modulation equations for the soliton parameters. We moreover prove a central limit theorem for the dispersive part of the solution, and investigate the asymptotic behavior in time of the limit process.