Long-Timescale Soliton Dynamics in the Korteweg-de Vries Equation with Multiplicative Translation-Invariant Noise

TL;DR

This paper investigates how solitons in the Korteweg-de Vries equation behave over long times under multiplicative noise, introducing stochastic models that accurately predict their amplitude and position dynamics.

Contribution

The study develops a stochastic framework for tracking soliton amplitude and position in noisy environments, providing new approximations and insights into their long-term behavior.

Findings

Stochastic models accurately predict soliton dynamics under noise.

Approximate formulas reveal the leading-order drift of soliton parameters.

Numerical evidence confirms the theoretical predictions.

Abstract

This paper studies the behavior of solitons in the Korteweg-de Vries equation under the influence of multiplicative noise. We introduce stochastic processes that track the amplitude and position of solitons based on a rescaled frame formulation and stability properties of the soliton family. We furthermore construct tractable approximations to the stochastic soliton amplitude and position which reveal their leading-order drift. We find that the statistical properties predicted by our method agree well with numerical evidence.