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Multi-fold Darboux transforms and interaction solutions of localized waves to a general vector mKdV equation

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A Correction to this article was published on 31 May 2023

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Abstract

In this paper, a new general vector mKdV equation associated with the \((m+n+1)\times (m+n+1)\) matrix spectral problem is proposed and its integrable reduced equation is derived. Based on the gauge transformations between the resulting Lax pairs and Riccati equations related to the spectral problem and auxiliary spectral problem, multi-fold Darboux transforms of the general vector mKdV equation and its integrable reduced equation are constructed, from which an algebraic algorithm for solving the general vector mKdV equation and its integrable reduced equation is given. As an illustrative example of the application of the Darboux transform, one obtains localized wave solutions of the integrable reduced equation such as solitons, soliton molecules, breathers and rogue waves. It is important and interesting that the integrable reduced equation has three new characteristics: (i) it has a two-atom soliton molecule and the two atoms overlap so well that it has only one crest, but it behaves like a two-atom soliton when it interacts with a third solitary wave; (ii) the interaction of its two waves may cause singular waves; and (iii) its analytical solution can be obtained from its singular seed solutions. In addition, the interaction dynamics of various localized wave solutions is analyzed by choosing appropriate parameters.

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Funding

This paper is supported by National Natural Science Foundation of China (Grant Nos. 12001496, 11931017, 11871440).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan, 450001, China

    Ruomeng Li & Yihao Li

  2. School of Computer and Artificial Intelligence, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan, 450001, China

    Jingru Geng

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  1. Ruomeng Li
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  2. Yihao Li
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Li, R., Li, Y. & Geng, J. Multi-fold Darboux transforms and interaction solutions of localized waves to a general vector mKdV equation. Nonlinear Dyn 111, 12525–12540 (2023). https://doi.org/10.1007/s11071-023-08482-y

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