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Numerical Methods for Some Nonlinear Schrödinger Equations in Soliton Management

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Abstract

In this work, we consider the numerical solutions of a dispersion-managed nonlinear Schrödinger equation (DM-NLS) and a nonlinearity-managed NLS equation (NM-NLS). The two equations arise from the soliton managements in optics and matter waves, and they involve temporal discontinuous coefficients with possible frequent jumps and stiffness which cause numerical difficulties. We analyze to see the order reduction problems of some popular traditional methods, and then we propose a class of exponential-type dispersion-map integrators for both DM-NLS and NM-NLS. The proposed methods are explicit, efficient under Fourier pseudospectral discretizations and second order accurate in time regardless the jumps/jump-period in the dispersion map. The extension to the fast & strong management regime of DM-NLS is made with uniform accuracy.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China 12271413, 11901440, the National Key Research and Development Program of China (No. 2020YFA0714200) and the Natural Science Foundation of Hubei Province No. 2019CFA007.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Ying He

  2. School of Mathematics and Statistics and Computational Sciences Hubei Key Laboratory, Wuhan University, Wuhan, 430072, China

    Xiaofei Zhao

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  1. Ying He
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He, Y., Zhao, X. Numerical Methods for Some Nonlinear Schrödinger Equations in Soliton Management. J Sci Comput 95, 61 (2023). https://doi.org/10.1007/s10915-023-02181-x

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