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Averaging principle for the higher order nonlinear Schrödinger equation with a random fast oscillation. (English) Zbl 1394.35473
Summary: This work concerns the problem associated with averaging principle for a higher order nonlinear Schrödinger equation perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. This model can be translated into a multiscale stochastic partial differential equations. Stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time-scales. To be more precise, under suitable conditions, we prove that there is a limit process in which the fast varying process is averaged out and the limit process which takes the form of the higher order nonlinear Schrödinger equation is an average with respect to the stationary measure of the fast varying process. Finally, by using the Khasminskii technique we can obtain the rate of strong convergence for the slow component towards the solution of the averaged equation, and as a consequence, the system can be reduced to a single higher order nonlinear Schrödinger equation with a modified coefficient.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q53 KdV equations (Korteweg-de Vries equations) 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 70K65 Averaging of perturbations for nonlinear problems in mechanics 70K70 Systems with slow and fast motions for nonlinear problems in mechanics 35R60 PDEs with randomness, stochastic partial differential equations
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