Research Article

Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Gautier, Éric

CREST-INSEE, URA D2200, 3 avenue Pierre Larousse, 92240 Malakoff, France.

IRMAR, UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; eric.gautier@bretagne.ens-cachan.fr

Abstract

Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.

(Received May 25 2004)

(Revised December 15 2004)

(Online publication November 15 2005)

Key Words:

• Large deviations;
• stochastic partial differential equations;
• nonlinear Schrödinger equations;
• white noise;
• projective limit;
• support theorem;
• blow-up;
• solitary waves.

Mathematics Subject Classification:

• 35Q51;
• 35Q55;
• 60F10;
• 60H15