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Bäcklund transformation, Lax pair and dynamic behaviour of exact solutions for a (\(3+1\))-dimensional nonlinear equation

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Abstract

In this article, we focuss primarily on a (\(3+1\))-dimensional nonlinear equation in fluids and the dynamic behaviour of the lump solution and new interaction solutions of the equation. First, the bilinear form of the equation is investigated using the Bell polynomials and the Hirota bilinear method. The Bell-polynomial-type Bäcklund transformation of the equation and the corresponding Lax pair are obtained by using the transformation technique. Based on the obtained bilinear form, we use the positive quadratic function method to obtain the lump solutions of the equation and study its dynamic behaviour in detail. On the other hand, we constructed two types of interaction solutions, lump-soliton and lump-periodic, and combined the images to vividly show the interaction phenomenon. Furthermore, the homoclinic breather solutions of the equation are constructed using the homoclinic test method. Finally, we hope that the solutions we obtained can explain some nonlinear phenomena in the fluid mechanics and shallow water wave fields.

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Acknowledgements

This project was supported by funding of Visual Computing and Virtual Reality Key Laboratory of Sichuan Province (Grant No. SCVCVR2023.12VS), Scientific Research Foundation of the Education Department of Sichuan Province, China (Grant No. 15ZB0362) and Scientific Research Foundation of Engineering & Technical College of Chengdu University of Technology (Grant No. C122022022).

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Authors and Affiliations

  1. The Engineering and Technical College of Chengdu University of Technology, Leshan, 614000, China

    Zhimin Ma, Binji Wang, Xukun Liu & Yuanlin Liu

  2. Visual Computing and Virtual Reality Key Laboratory of Sichuan Province, Sichuan Normal University, Chengdu, 610068, China

    Zhimin Ma

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  1. Zhimin Ma
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  2. Binji Wang
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  3. Xukun Liu
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  4. Yuanlin Liu
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Correspondence to Zhimin Ma.

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Ma, Z., Wang, B., Liu, X. et al. Bäcklund transformation, Lax pair and dynamic behaviour of exact solutions for a (\(3+1\))-dimensional nonlinear equation. Pramana - J Phys 98, 24 (2024). https://doi.org/10.1007/s12043-023-02721-y

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