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Riemann–Hilbert approach and soliton classification for a nonlocal integrable nonlinear Schrödinger equation of reverse-time type

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Abstract

In this paper, a Riemann–Hilbert (RH) approach is reported for a physically meaningful nonlocal integrable nonlinear Schrödinger equation of reverse-time type, which is connected with a special initial problem of the Manakov system. In this RH approach, the spectral analysis is performed from the x-part of the Lax pair to formulate the desired RH problem. Using the symmetry properties of the potential matrix, the zero structure of the RH problem is investigated in detail. The obtained results mainly comprise (i) the symmetry relations of the scattering data are successfully found, (ii) the general multi-soliton solutions are obtained in the reflectionless cases and classified into three categories according to three types of zeros of the RH problem, and (iii) the long-time behaviors of solutions are shown by solving the RH problem in the reflection cases. Additionally, to show the remarkable features of the obtained multi-soliton solutions, some special soliton dynamics are explored and graphically illustrated using Mathematica. Moreover, the multi-soliton solutions obtained for the nonlocal integrable nonlinear Schrödinger equation can be used to construct solutions of the Manakov system with the specific initial condition.

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Acknowledgements

The author expresses sincere thanks 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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  1. School of Science, Zhengzhou University of Aeronautics, Zhengzhou 450046, Henan, People’s Republic of China

    Jianping Wu

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Wu, J. Riemann–Hilbert approach and soliton classification for a nonlocal integrable nonlinear Schrödinger equation of reverse-time type. Nonlinear Dyn 107, 1127–1139 (2022). https://doi.org/10.1007/s11071-021-07005-x

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