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Bilinear Form, Bilinear Bäcklund Transformations, Breather and Periodic-Wave Solutions for a (2+1)-Dimensional Shallow Water Equation with the Time-Dependent Coefficients

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Abstract

Shallow water waves are seen in geophysical fluid dynamics, oceanography, coastal engineering and atmospheric science. In this paper, to describe the shallow water waves, we investigate a (2+1)-dimensional shallow water equation with the time-dependent coefficients via symbolic computation. Based on the Hirota method and Painlevé integrable conditions in the existing literature, the bilinear form for that equation is hereby constructed. Via the bilinear form and exchange formulae, we build three bilinear Bäcklund transformations with certain soliton-like solutions. Via the extended homoclinic test approach, we derive the breather solutions and their asymptotic behaviors. We graphically show the breather waves. Periodic-wave solutions are worked out via the Hirota-Riemann method, and graphically displayed. Relation between the periodic-wave solutions and one-soliton solutions is discussed.

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Notes

  1. Reference [34] has researched the Painlevé integrable conditions, i.e., Conditions (7), and gained some real- and complex-soliton solutions for Eq. (1).

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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. 11772017, 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

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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

    Chun-Hui Feng, Bo Tian, Dan-Yu Yang & Xiao-Tian Gao

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  1. Chun-Hui Feng
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C.-H.F.: Writing-original draft, writing-review & editing. B.T.: Supervision, writing-review. D.-Y.Y.: Validation, writing-review & editing. X.-T.G.: Validation, writing-review & editing.

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Correspondence to Chun-Hui Feng or Bo Tian.

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Feng, CH., Tian, B., Yang, DY. et al. Bilinear Form, Bilinear Bäcklund Transformations, Breather and Periodic-Wave Solutions for a (2+1)-Dimensional Shallow Water Equation with the Time-Dependent Coefficients. Qual. Theory Dyn. Syst. 22, 147 (2023). https://doi.org/10.1007/s12346-023-00813-z

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