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Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations

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Abstract

A study of high-order solitons in three nonlocal nonlinear Schrödinger equations is presented. These include the \(\mathcal {PT}\)-symmetric, reverse-time, and reverse-space-time nonlocal nonlinear Schrödinger equations. General high-order solitons in three different equations are derived from the same Riemann–Hilbert solutions of the AKNS hierarchy, except for the difference in the corresponding symmetry relations on the “perturbed” scattering data. Dynamics of general high-order solitons in these equations is further analyzed. It is shown that the high-order fundamental-soliton is moving on several different trajectories in nearly equal velocities, and they can be nonsingular or repeatedly collapsing, depending on the choices of the parameters. It is also shown that the high-order multi-solitons could have more complicated wave structures and behave very differently from high-order fundamental-solitons. More interestingly, via the combinations of different size of block matrix in the Riemann–Hilbert solutions, high-order hybrid-pattern solitons are found, which describe the nonlinear interaction between several types of solitons.

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Acknowledgements

This project is supported by the Global Change Research Program of China (No. 2015CB953904), National Natural Science Foundation of China (Nos. 11675054 and 11435005), and Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (No. ZF1213).

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China

    Bo Yang & Yong Chen

  2. Department of Physics, Zhejiang Normal University, Jinhua, 321004, China

    Yong Chen

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  1. Bo Yang
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Correspondence to Yong Chen.

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Yang, B., Chen, Y. Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations. Nonlinear Dyn 94, 489–502 (2018). https://doi.org/10.1007/s11071-018-4373-0

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