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Interaction of wave structure in the \(\mathcal{P}\mathcal{T}\)-symmetric \((3\,+\,1)\)-dimensional nonlocal Mel’nikov equation and their applications

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Abstract

The \((3+1)\)-dimensional [\((3+1)\)-d] Mel’nikov equation describes an interaction between long-wave and short-wave packets in three-spatial dimensions, which is the coupling of \((3+1)\)-d Kadomtsev–Petviashvili [KP] equation and nonlinear Schrödinger [NLS] equation. In this paper, its \(\mathcal{P}\mathcal{T}\)-symmetric version is introduced, and we call it \((3+1)\)-d nonlocal Mel’nikov equation. General soliton solutions, including crossed soliton and parallel soliton, are presented in terms of KP hierarchy reduction method. Furthermore, the semi-rational solutions consisting of lumps and solitons are also constructed. These semi-rational solutions are elastic collision. However, the corresponding semi-rational solution of most nonlocal two-dimensional systems describes the inelastic collision between the lump waves and the solitons under the same conditions. Additionally, a new way to get the rational solution of Mel’nikov equation is given by reducing the semi-rational solution of the \((3+1)\)-d nonlocal Mel’nikov equation. These novel dynamics have never been reported in \((3+1)\)-d nonlocal systems. Moreover, it broadens our research field and inspires us to explore the mysteries of higher-dimensional nonlocal systems.

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Acknowledgements

The authors would like to thank Prof. Jingsong He of Shenzhen University for his fruitful suggestions. This work is supported by the NSF of China under Grant No. 12161048. Doctoral Research Foundation of Jiangxi University of Chinese Medicine, China (Grant No. 2021WBZR007) and Jiangxi University of Chinese Medicine Science and Technology Innovation Team Development Program, China (Grant No. CXTD22015).

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Authors and Affiliations

  1. School of Mathematics and Science, Nanyang Institute of Technology, Nanyang, 473004, Henan, People’s Republic of China

    Yulei Cao & Hao Tian

  2. Department of Mathematics, Saint Xavier University, Chicago, IL, 60655, USA

    Abdul-Majid Wazwaz

  3. College of Computer, Jiangxi University of Chinese Medicine, Nanchang, 330004, Jiangxi, People’s Republic of China

    Jian-Guo Liu

  4. Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou, 510631, People’s Republic of China

    Zhao Zhang

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  1. Yulei Cao
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  3. Abdul-Majid Wazwaz
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  4. Jian-Guo Liu
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Correspondence to Jian-Guo Liu.

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Cao, Y., Tian, H., Wazwaz, AM. et al. Interaction of wave structure in the \(\mathcal{P}\mathcal{T}\)-symmetric \((3\,+\,1)\)-dimensional nonlocal Mel’nikov equation and their applications. Z. Angew. Math. Phys. 74, 49 (2023). https://doi.org/10.1007/s00033-023-01945-7

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  • DOI: https://doi.org/10.1007/s00033-023-01945-7

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