Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications. (English) Zbl 1136.60346
Summary: 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.
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35Q51 | Soliton equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 60F10 | Large deviations |
Keywords:
Large deviations; stochastic partial differential equations; nonlinear Schrödinger equations; white noise; projective limit; support theorem; blow-up; solitary wavesReferences:
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