Random modulation of solitons for the stochastic Korteweg-de Vries equation

We study the asymptotic behavior of the solution of a Korteweg-de Vries equation with an additive noise whose amplitude ε tends to zero. The noise is white in time and correlated in space and the initial state of the solution is a soliton solution of the unperturbed Korteweg-de Vries equation. We prove that up to times of the order of 1 / ε², the solution decomposes into the sum of a randomly modulated soliton, and a small remainder, and we derive the equations for the modulation parameters. We prove in addition that the first order part of the remainder converges, as ε tends to zero, to a Gaussian process, which satisfies an additively perturbed linear equation.