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International audienceWe 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

Small noise asymptotic of the timing jitter in soliton transmission