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Double-loop multivalue solitary waves and their collisions for a 2D KdV equation

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Abstract

The Korteweg–de Vries (KdV) equation can describe a weakly nonlinear long wave when its phase speed arrives a wave maximum with the infinite length in shallow water. In this paper, the investigation of a two-dimensional (2D) KdV equation is conducted using the double-function method. A unified solution for separating the variables is given based on the projective Riccati equation in the double-function method and its unified exponential-form solution, which covers many familiar solutions including \(\sinh ,\cosh ,\text {sech},\tanh ,\text {csch},\coth ,\sec ,\tan ,\csc ,\cot \) solutions in previous literatures. From this exponential-form solution for the separation of the variables, the imperfect elastic collision between bell-shaped and peak-shaped double-loop semi-folded solitary waves, imperfect elastic collision between peak-shaped double-loop semi-folded and double-loop full-folded solitary waves and perfectly inelastic collision between bell-shaped and peak-shaped double-loop semi-folded solitary waves are graphically and analytically discussed. Moreover, the collision properties between solitary waves are quantificationally analysed by the asymptotic analysis. Phase shifts and their difference values of collisions between double-loop multivalue solitary waves are analytically presented.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11975197).

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Authors and Affiliations

  1. School of Electronics Information, Communication University of Zhejiang, Hangzhou, 310018, People’s Republic of China

    Yi-Xiang Chen & Xiao Xiao

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  1. Yi-Xiang Chen
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  2. Xiao Xiao
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Correspondence to Yi-Xiang Chen.

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Chen, YX., Xiao, X. Double-loop multivalue solitary waves and their collisions for a 2D KdV equation. Pramana - J Phys 96, 202 (2022). https://doi.org/10.1007/s12043-022-02448-2

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  • DOI: https://doi.org/10.1007/s12043-022-02448-2

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