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Breather and Interaction Solutions for a \((3+1)\)-Dimensional Generalized Shallow Water Wave Equation

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Abstract

In this paper, we attempt to construct the breather and interaction solutions for a \((3+1)\)-dimensional generalized shallow water wave equation. Such equation has been utilized to describe the long water waves in an ocean, impoundment or estuary, and used in the tsunami predictions, river/tidal-wave/irrigation flow studies, as well as weather simulations. Via the Kadomtsev–Petviashvili hierarchy reduction method and taking the long-wave limit technique, we derive the breather and interaction solutions in terms of the Gramian, which have never been investigated before. Each interval of the breather possesses one peak and one valley, and the interaction between two breathers is elastic. When we take the long-wave limit in the breather solutions, interaction solutions are demonstrated. Furthermore, we investigate graphically the dynamical behaviors of interaction solutions and find that the interaction between lump and breather is also elastic.

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Acknowledgements

We would like to express our sincere thanks to the referees for their valuable comments. This work has been supported by the Fundamental Research Funds for the Central Universities of China and the Basic Scientific Research Project of Education Department of Liaoning Province under Grant No. LJKMZ20220370.

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  1. School of Science, Dalian Maritime University, Dalian, 116026, People’s Republic of China

    Yan Sun

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Sun, Y. Breather and Interaction Solutions for a \((3+1)\)-Dimensional Generalized Shallow Water Wave Equation. Qual. Theory Dyn. Syst. 22, 91 (2023). https://doi.org/10.1007/s12346-023-00793-0

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