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Averaging principle for Korteweg–de Vries equation with a random fast oscillation

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Abstract

This work concerns averaging principle for Korteweg–de Vries equation perturbed by a oscillating term which arises from the solution of a stochastic reaction–diffusion equation. This model can be translated into multiscale stochastic partial differential equations. Under some suitable conditions, we show that the slow component strongly converges to the solution of a single Korteweg–de Vries equation with a modified coefficient (averaged equation). To be more precise, by using the Khasminskii technique, we can obtain the strong convergence rate of the slow component towards the solution of the corresponding averaged equation, and as a consequence, the multiscale system can be reduced to the averaged equation.

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Acknowledgements

I would like to thank the referees and the editor for their careful comments and useful suggestions. 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.

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This work is supported by NSFC Grant (11601073).

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Gao, P. Averaging principle for Korteweg–de Vries equation with a random fast oscillation. Z. Angew. Math. Phys. 70, 123 (2019). https://doi.org/10.1007/s00033-019-1165-4

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  • DOI: https://doi.org/10.1007/s00033-019-1165-4

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