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Multi-lump formations from lump chains and plane solitons in the KP1 equation

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Abstract

We show that complex higher-order lump patterns can be constructed in two different ways within the Kadomtsev–Petviashvili (KP1) equation which describes nonlinear wave processes in media with positive dispersion. In the first approach, we start with solutions describing stationary moving higher-order lump chains. By degenerating these solutions, we obtain first coupled lump chains which reduce then to multi-lump bound states in the limit when the period of lump chains goes to infinity. In another approach, a skilful technique is exploited to derive multi-lump bound states directly from the N-soliton solution of the KP1 equation presented in the Hirota form. It is shown then that through the proper selection of soliton parameters, these higher-order solutions can be reduced to the various lump patterns, such as triangular, polygonal patterns, and so on. The suggested approaches can be extended to another one- and multi-dimensional integrable systems to derive complex bound states and rogue waves.

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Acknowledgements

This research is supported by the Natural Science Foundation of Guangdong Province of China (No. 2021A1515012214), the Science and Technology Program of Guangzhou (No. 2019050001), National Natural Science Foundation of China (No. 12175111) and K.C.Wong Magna Fund in Ningbo University. The authors sincerely thank Dr. Junchao Chen (Lishui University) for the discussions. Yu.A. Stepanyants acknowledges the funding provided by the Council of the grants of the President of the Russian Federation for the state support of Leading Scientific Schools of the Russian Federation (Project No. NSH–70.2022.1.5).

Funding

This study was funded by the Natural Science Foundation of Guangdong Province of China (No. 2021A1515012214) and the Science and Technology Program of Guangzhou (No. 2019050001).

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Authors and Affiliations

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

    Zhao Zhang & Qi Guo

  2. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xiangyu Yang

  3. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Biao Li

  4. School of Mathematics, Physics and Computer Sciences, University of Southern Queensland, Toowoomba, QLD, 4350, Australia

    Yury Stepanyants

  5. Department of Applied Mathematics, Nizhny Novgorod State Technical University, n.a. R.E. Alekseev, 24 Minin St., Nizhny Novgorod, Russia, 603950

    Yury Stepanyants

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  1. Zhao Zhang
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Correspondence to Qi Guo.

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Zhang, Z., Yang, X., Li, B. et al. Multi-lump formations from lump chains and plane solitons in the KP1 equation. Nonlinear Dyn 111, 1625–1642 (2023). https://doi.org/10.1007/s11071-022-07903-8

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