We use the inverse scattering transform and a diffusion approximation limittheorem to study the stability of soliton components of the solution of thenonlinear Schr\"{o}dinger and Korteweg-de Vries equations under randomperturbations of the initial conditions: for a wide class of rapidlyoscillating random perturbations this problem reduces to the study of acanonical system of stochastic differential equations which depends only on theintegrated covariance of the perturbation. We finally study the problem whenthe perturbation is weak, which allows us to analyze the stability of solitonsquantitatively.
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