Exit problems related to the persistence of solitons for the Korteweg-de Vries equation with small noise

TL;DR

This paper analyzes the exit times of solitons in the stochastic Korteweg-de Vries equation, providing precise probabilistic bounds and insights into soliton persistence under small noise perturbations.

Contribution

It offers the first exponential bounds for exit times from neighborhoods of solitons in the stochastic KdV equation, clarifying the influence of noise and secular modes.

Findings

Exponential bounds for exit probabilities in small noise limit.

Quantification of secular modes' impact on soliton persistence.

Time scale matches the theoretical prediction for soliton stability.

Abstract

We consider two exit problems for the Korteweg-de Vries equation perturbed by an additive white in time and colored in space noise of amplitude a. The initial datum gives rise to a soliton when a=0. It has been proved recently that the solution remains in a neighborhood of a randomly modulated soliton for times at least of the order of a^{-2}. We prove exponential upper and lower bounds for the small noise limit of the probability that the exit time from a neighborhood of this randomly modulated soliton is less than T, of the same order in a and T. We obtain that the time scale is exactly the right one. We also study the similar probability for the exit from a neighborhood of the deterministic soliton solution. We are able to quantify the gain of eliminating the secular modes to better describe the persistence of the soliton.