The Annals of Applied Probability

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

Ennio Fedrizzi

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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 Schrödinger 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.

Article information

Ann. Appl. Probab. Volume 24, Number 2 (2014), 616-651.

First available in Project Euclid: 10 March 2014

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Zentralblatt MATH identifier

Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case) 35C08: Soliton solutions
Secondary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]

Diffusion approximation limit theorem random perturbation of initial conditions solitons NLS equation KdV equation


Fedrizzi, Ennio. Stability of solitons under rapidly oscillating random perturbations of the initial conditions. Ann. Appl. Probab. 24 (2014), no. 2, 616--651. doi:10.1214/13-AAP931.

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