Stability of solitons under rapidly oscillating random perturbations of the initial conditions

Ennio Fedrizzi (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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.

Mots clés: Diffusion approximation limit theorem; random perturbation of initial conditions; solitons; NLS equation; KdV equation

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Ennio Fedrizzi. Stability of solitons under rapidly oscillating random perturbations of the initial conditions. 2011. ⟨hal-01017943⟩