Stability of Stochastically Forced Solitons in the Korteweg-de Vries Equation
Rik W. S. Westdorp, Hermen Jan Hupkes
TL;DR
Stability of stochastically forced solitons in the KdV equation is established by a rigorous modulation framework in weighted spaces. The authors derive global and local modulation systems for the soliton amplitude and phase, and prove high-probability bounds for the remainder and for the reduced amplitude dynamics, incorporating both deterministic forcing and space-colored noise. They obtain explicit exit-time bounds and show that the effective amplitude equation governs the leading stochastic modulation, even when forcing induces sizable amplitude changes. The results provide a rigorous justification of soliton robustness under realistic stochastic forcing and yield practical modulation equations for predicting soliton behavior.
Abstract
We study the stability and dynamics of solitons in the Korteweg-de Vries (KdV) equation in the presence of noise and deterministic forcing. The noise is space-dependent and statistically translation-invariant. We show that, for small forcing, solitons remain close to the family of traveling waves in a weighted Sobolev norm, with high probability. We study the effective dynamics of the soliton amplitude and position via their variational phase, for which we derive explicit modulation equations. The stability result holds on a time scale where the deterministic forcing induces significant amplitude modulation.