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Symplectic simulation of dark solitons motion for nonlinear Schrödinger equation

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Abstract

In this paper, we study symplectic simulation of dark solitons motion of nonlinear Schrödinger equation (NLSE). The Ablowitz-Ladik model (A-L model) of NLSE can be expressed as a non-canonical Hamiltonian system. By using splitting technique, we construct explicit splitting K-symplectic methods for the A-L model. On the other hand, the A-L model can be transformed into a canonical system and standard symplectic methods can be employed to perform numerical simulation. A second order K-symplectic method and a second order symplectic method are employed to simulate one dark soliton and two dark solitons motion for the A-L model and its canonicalized system respectively. By comparing with a third-order non-symplectic Runge-Kutta method, we show the superiorities of the two symplectic methods in long-term tracking the motion of dark solitons and preserving the invariants. We also compare the CPU times of K-symplectic methods and standard symplectic methods and show that the former ones are more efficient. The energy-preserving scheme is also applied for non-canonical Hamiltonian systems. The numerical results demonstrate that the K-symplectic methods can nearly preserve the energy, the discrete invariants of A-L model and conserved quantities of NLSE, but the energy-preserving scheme can only exactly preserve the energy.

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Acknowledgements

We are grateful to Zhaoqi Zhou for useful discussion in energy-preserving method.

Funding

This research is supported by the National Natural Science Foundation of China (Grant No. 11771438). Beibei Zhu was supported by the National Center for Mathematics and Interdisciplinary Sciences, CAS.

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Authors and Affiliations

  1. National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Beibei Zhu

  2. LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Yifa Tang & Yihao Zhang

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Yifa Tang & Yihao Zhang

  4. School of Science, Beijing Jiaotong University, Beijing, 100044, China

    Ruili Zhang

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  1. Beibei Zhu
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  2. Yifa Tang
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  3. Ruili Zhang
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  4. Yihao Zhang
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Correspondence to Beibei Zhu.

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Zhu, B., Tang, Y., Zhang, R. et al. Symplectic simulation of dark solitons motion for nonlinear Schrödinger equation. Numer Algor 81, 1485–1503 (2019). https://doi.org/10.1007/s11075-019-00708-8

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  • DOI: https://doi.org/10.1007/s11075-019-00708-8

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