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Looking at an open sea via a generalized \((2+1)\)-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons

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Abstract

Oceanic water-wave studies are attractive. Hereby, with a view to modelling the nonlinear and dispersive long gravity waves in two horizontal directions on the shallow water of an open sea or a wide channel of finite depth, we inquire into a generalized (2+1)-dimensional dispersive long-wave system, by virtue of the scaling transformations we hereby obtain, Bell polynomials and symbolic computation. As for the horizontal velocity and wave elevation above the undisturbed water surface, we find two branches of the hetero-Bäcklund transformations, two branches of the bilinear forms and two branches of the N-soliton solutions, with N as a positive integer. What we have obtained relies on the coefficients in the system.

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Notes

  1. More on symbolic computation can be seen. e.g., in Refs. [46,47,48,49,50,51,52,53,54,55,56,57].

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Acknowledgements

We express our sincere thanks to the editors and reviewers for their valuable comments. This work was supported by the National Nature Science Foundation of China under Grant No. 11871116 and Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11. XYG also thanks the National Scholarship for Doctoral Students of China and BUPT Innovation and Entrepreneurship Support Program, Beijing University of Posts and Telecommunications.

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Authors and Affiliations

  1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Xin-Yi Gao, Yong-Jiang Guo & Wen-Rui Shan

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  1. Xin-Yi Gao
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  2. Yong-Jiang Guo
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  3. Wen-Rui Shan
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Correspondence to Yong-Jiang Guo or Wen-Rui Shan.

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Gao, XY., Guo, YJ. & Shan, WR. Looking at an open sea via a generalized \((2+1)\)-dimensional dispersive long-wave system for the shallow water: scaling transformations, hetero-Bäcklund transformations, bilinear forms and N solitons. Eur. Phys. J. Plus 136, 893 (2021). https://doi.org/10.1140/epjp/s13360-021-01773-6

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