Exit Times and Persistence of Solitons for a Stochastic Korteweg-de Vries Equation

The Korteweg-de Vries equation is a model of nonlinear shallow water long waves of small amplitude that admit soliton solutions. Solitons are a family of solutions which are progressive localized waves that propagate with constant speed and shape. These waves are stable in many ways against perturbations or interactions. We consider random perturbations by an additive noise of small amplitude. It is common in Physics to approximate the solution in the presence of noise, corresponding to an initial datum generating a soliton in the deterministic system, by a randomly modulated soliton (the parameters of the soliton fluctuate randomly). The validity of such an approximation has been proved by A. de Bouard and A. Debussche. We present results obtained in a joint work with A. de Bouard where we study in more details the exit time from a neighborhood of the soliton and randomly modulated soliton and obtain the scaling in terms of the amplitude of the noise for each approximation. This allows to quantify the gain of an approximation of the form of a randomly modulated soliton in describing the persistence of solitons.