Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation

Abstract

The nonlocal defocusing nonlinear Schrödinger (ND-NLS) equation is comparatively studied via the Riemann–Hilbert approach. Firstly, via spectral analysis, the spectral structure of the ND-NLS equation is investigated, which is different to those of the other three NLS-type equations, i.e., the local focusing nonlinear Schrödinger (LF-NLS) equation, the local defocusing nonlinear Schrödinger (LD-NLS) equation and the nonlocal focusing nonlinear Schrödinger (NF-NLS) equation. Secondly, by solving the Riemann–Hilbert problem corresponding to the reflectionless cases, multi-soliton solutions are obtained for the ND-NLS equation. Thirdly, we prove that, if parameters are suitably chosen, the multi-soliton solutions of the ND-NLS equation can be reduced to those of the LF-NLS equation and the LD-NLS equation, respectively. Fourthly, the multi-soliton solutions of the ND-NLS equation are demonstrated to possess repeated singularities generally, but they can also remain analytic for appropriate soliton parameters. Moreover, the multi-soliton dynamics are graphically illustrated using Mathematica symbolic computations. These results show that the solution structure and the nonlinear dynamics in the ND-NLS equation are rather different from those of the LF-NLS equation, the LD-NLS equation and the NF-NLS equation.

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Acknowledgements

The author is very grateful to the editor and the anonymous referees for their valuable suggestions. The author would also like to thank the support by the Collaborative Innovation Center for Aviation Economy Development of Henan Province.

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Correspondence to Jianping Wu.

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Wu, J. Riemann–Hilbert approach and nonlinear dynamics in the nonlocal defocusing nonlinear Schrödinger equation. Eur. Phys. J. Plus 135, 523 (2020). https://doi.org/10.1140/epjp/s13360-020-00348-1

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