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Dynamics of General Soliton and Rational Solutions in the \((3+1)\)-Dimensional Nonlocal Mel’nikov Equation with Non-zero Background

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Abstract

Employing the KP reduction approach, the primary goal of this research work is to investigate the soliton and rational solutions of the \((3+1)\)-dimensional nonlocal Mel’nikov equation with non-zero background. The solutions presented are all \(N\times N\) Gram determinants. In contrast to the previous exact solutions of the nonlocal model obtained by the KP reduction method, we introduce two types of parameter constraints into the \(\tau \) function. This leads to the appearance of rational solutions and soliton (breather) solutions against a background of periodic wave. In particular, the soliton types we obtained are dark soliton, antidark soliton, breather, periodic wave and degenerate soliton. Furthermore, it has been discovered that lumps can appear in odd or even numbers in two backgrounds, which is a novel finding. The dynamic behavior of all solutions has been comprehensively analyzed.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371326, 11975145 and 12271488).

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China

    Xiaolin Yang, Yi Zhang & Wenjing Li

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  1. Xiaolin Yang
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  2. Yi Zhang
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X.Y. write the main manuscript text , Y.Z. reviewed the manuscript and all authors reviewed the manuscript.

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Correspondence to Yi Zhang.

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Yang, X., Zhang, Y. & Li, W. Dynamics of General Soliton and Rational Solutions in the \((3+1)\)-Dimensional Nonlocal Mel’nikov Equation with Non-zero Background. Qual. Theory Dyn. Syst. 23, 211 (2024). https://doi.org/10.1007/s12346-024-01068-y

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