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Symbolically Computing the Shallow Water via a (2+1)-Dimensional Generalized Modified Dispersive Water-Wave System: Similarity Reductions, Scaling and Hetero-Bäcklund Transformations

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Abstract

For the water waves, people consider some dispersive systems. Describing the nonlinear and dispersive long gravity waves travelling along two horizontal directions in the shallow water of uniform depth, we now symbolically compute a (2+1)-dimensional generalized modified dispersive water-wave system. With respect to the height of the water surface and horizontal velocity of the water wave, with symbolic computation, we work out (1) a set of the scaling transformations, (2) a set of the hetero-Bäcklund transformations, from that system to a known linear partial differential equation, and (3) four sets of the similarity reductions, each of which is from that system to a known ordinary differential equation. We pay attention that our hetero-Bäcklund transformations and similarity reductions rely on the coefficients in that system.

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11871116 and 11772017, and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11. X. Y. Gao also thanks the National Scholarship for Doctoral Students of China.

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  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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Gao, XY., Guo, YJ. & Shan, WR. Symbolically Computing the Shallow Water via a (2+1)-Dimensional Generalized Modified Dispersive Water-Wave System: Similarity Reductions, Scaling and Hetero-Bäcklund Transformations. Qual. Theory Dyn. Syst. 22, 17 (2023). https://doi.org/10.1007/s12346-022-00684-w

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