Small noise asymptotic of the timing jitter in soliton transmission

Arnaud Debussche (1) , Eric Gautier (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider random perturbations of the focusing cubic one dimensional nonlinear Schrödinger equation. The noises, either additive or multiplicative, are white in time and colored in space. In the additive case, a "white noise limit" is considered. We study the small noise asymptotic of the tails of the center and mass of a pulse at a fixed coordinate when the initial datum is null or a soliton profile. Our main tools are large deviation results at the level of paths. Upper and lower bounds are obtained from bounds for the optimal control problems derived from the rate function of the large deviation principles. Our results are in perfect agreement with several results from physics.These results had been obtained with arguments which seem difficult to fully justify mathematically. Some results are new.

Mots clés: Large deviations; stochastic partial differential equations; nonlinear Schrödinger equation; solitons

Domaines: Probabilités [math.PR]

Arnaud Debussche, Eric Gautier. Small noise asymptotic of the timing jitter in soliton transmission. The Annals of Applied Probability, 2008, 18 (1), pp.178-208. ⟨10.1214/07-AAP449⟩. ⟨hal-00019026⟩