Long-timescale soliton dynamics in the Korteweg–de Vries equation with multiplicative translation-invariant noise

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.