Stability of solitons under rapidly oscillating random perturbations of the initial conditions

TL;DR

This paper investigates the stability of solitons in nonlinear Schrödinger and Korteweg-de Vries equations under rapidly oscillating random initial perturbations, using inverse scattering and stochastic analysis to derive quantitative stability results.

Contribution

It introduces a novel approach combining inverse scattering and diffusion approximation to analyze soliton stability under complex random perturbations.

Findings

Stability reduces to a stochastic differential equation system depending on covariance.

Quantitative stability analysis for weak perturbations.

Framework applicable to a wide class of oscillating random perturbations.

Abstract

We use the inverse scattering transform and a diffusion approximation limit theorem to study the stability of soliton components of the solution of the nonlinear Schr\"{o}dinger and Korteweg-de Vries equations under random perturbations of the initial conditions: for a wide class of rapidly oscillating random perturbations this problem reduces to the study of a canonical system of stochastic differential equations which depends only on the integrated covariance of the perturbation. We finally study the problem when the perturbation is weak, which allows us to analyze the stability of solitons quantitatively.