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Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation

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Abstract

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.

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Acknowledgements

I sincerely thank Professor Yong Li for many useful suggestions and help.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, People’s Republic of China

    Peng Gao

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  1. Peng Gao
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Correspondence to Peng Gao.

Additional information

This work is supported by NSFC Grant (11601073), NSFC Grant (11701078), China Postdoctoral Science Foundation (2017M611292) and the Fundamental Research Funds for the Central Universities (2412017QD002).

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Gao, P. Averaging Principle for the Higher Order Nonlinear Schrödinger Equation with a Random Fast Oscillation. J Stat Phys 171, 897–926 (2018). https://doi.org/10.1007/s10955-018-2048-3

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  • DOI: https://doi.org/10.1007/s10955-018-2048-3

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