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Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations

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Abstract

In this paper, the Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup (WBK) equations are studied, which can help us increase the diversity of solutions and get more new phenomena. The integrability of WBK equations is discussed by Painlevé analysis method, and the Bäcklund transformation between WBK equations and linear equation is obtained by using the truncated expansion. According to the obtained Bäcklund transformation, some superposition wave solutions of WBK equations are constructed. We analyze the properties of the solutions with the help of the symbolic calculation system Mathematica. Through analysis, there is a mathematical relationship between the number of waves and the number of superposition functions, and in the same case, u and v show similar contours in the solutions of the equations. Furthermore, by superimposing the solutions of the transformed linear equation and introducing arbitrary functions, we obtain superposition wave solutions of the nonlinear evolution equations. Also, it has been observed that the results obtained in this work have been presented for the first time.

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Acknowledgements

The authors deeply appreciate the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further. This work is supported by the National Natural Science Foundation of China (Grant No.11361040), the Natural Science Foundation of Inner Mongolia Autonomous Region, China (Grant No.2020LH01008), and the Graduate Students’s Scientific Research Innovation Fund Program of Inner Mongolia Normal University, China (Grant No.CXJJS20089).

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Authors and Affiliations

  1. College of Mathematics Science, Inner Mongolia Normal University, Huhhot, 010022, People’s Republic of China

    Lulu Fan & Taogetusang Bao

  2. Center for Applied Mathematics Inner Mongolia, Huhhot, 010022, People’s Republic of China

    Lulu Fan & Taogetusang Bao

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  1. Lulu Fan
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  2. Taogetusang Bao
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Correspondence to Taogetusang Bao.

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Fan, L., Bao, T. Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations. Nonlinear Dyn 109, 3091–3100 (2022). https://doi.org/10.1007/s11071-022-07605-1

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