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Bilinear Bäcklund transformation, soliton and periodic wave solutions for a \(\varvec{(3 + 1)}\)-dimensional variable-coefficient generalized shallow water wave equation

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Abstract

Under investigation in this paper is a \((3 + 1)\)-dimensional variable-coefficient generalized shallow water wave equation. Bilinear forms, Bäcklund transformation and Lax pair are obtained based on the Bell polynomials and symbolic computation. One-, two- and three-soliton solutions are derived via the Hirota method. One-periodic wave solutions are obtained via the Hirota–Riemann method. Discussions indicate that the one-periodic wave solutions approach to the one-soliton solutions when \(\varTheta \rightarrow 0\). Propagation and interaction of the soliton solutions have been discussed graphically. We find that not the soliton amplitudes, but the velocities are related to the variable coefficients \(\delta _{1}(t)\) and \(\delta _{2}(t)\). Phase shifts of the two-soliton solutions are the only differences to the superposition of two one-soliton solutions, so the amplitudes of the two-soliton solutions are equal to the sum of the corresponding two one-soliton solutions.

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Acknowledgements

We express our sincere thanks to the editors, reviewers and members of our discussion group for their valuable suggestions. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, and by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications).

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Authors and Affiliations

  1. Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Qian-Min Huang, Yi-Tian Gao, Shu-Liang Jia, Ya-Le Wang & Gao-Fu Deng

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  1. Qian-Min Huang
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  2. Yi-Tian Gao
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  3. Shu-Liang Jia
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  4. Ya-Le Wang
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Correspondence to Yi-Tian Gao.

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Huang, QM., Gao, YT., Jia, SL. et al. Bilinear Bäcklund transformation, soliton and periodic wave solutions for a \(\varvec{(3 + 1)}\)-dimensional variable-coefficient generalized shallow water wave equation. Nonlinear Dyn 87, 2529–2540 (2017). https://doi.org/10.1007/s11071-016-3209-z

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