We use the inverse scattering transform and a diffusion approximation limit
theorem to study the stability of soliton components of the solution of the
nonlinear Schr\"{o}dinger and Korteweg-de Vries equations under random
perturbations of the initial conditions: for a wide class of rapidly
oscillating random perturbations this problem reduces to the study of a
canonical system of stochastic differential equations which depends only on the
integrated covariance of the perturbation. We finally study the problem when
the perturbation is weak, which allows us to analyze the stability of solitons
quantitatively.Comment: Published in at http://dx.doi.org/10.1214/13-AAP931 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org