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On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

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Abstract

Describing the long waves in shallow water, a generalized Broer-Kaup-Kupershmidt system is investigated in this paper. With respect to the horizontal velocity of the water wave and the height of the water surface, we use symbolic computation to build up (A) a scaling transformation, (B) a set of the hetero-Bäcklund transformations, from that generalized system to a known linear partial differential equation, as well as (C) two sets of the similarity reductions, each of which from that generalized system to a known ordinary differential equation. Our results depend on all the shallow-water coefficients for that generalized system.

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Notes

  1. similar to those in Refs. [29, 30]

  2. similar to those in Refs. [35,36,37,38,39,40,41]

  3. There exist three freedoms in Remark 3, as seen in Ref. [35].

  4. More related symbolic-computation studies on other nonlinear evolution equations have been reported, i.e., in Refs. [44,45,46,47,48,49,50,51].

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Acknowledgements

We express our sincere thanks to the Editors and Reviewers for their valuable comments.

Funding

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. 20S19XD-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. On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water. Nonlinear Dyn 111, 9431–9437 (2023). https://doi.org/10.1007/s11071-023-08299-9

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