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Bäcklund transformations and soliton solutions for a (2 + 1)-dimensional Korteweg–de Vries-type equation in water waves

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Abstract

Under investigation in this paper is a \((2+1)\)-dimensional Korteweg–de Vries-type equation, which can describe the propagation of nonlinear waves such as the shallow-water waves, surface and internal waves. By virtue of the Bell polynomials, symbolic computation and auxiliary independent variable, the bilinear forms, Bäcklund transformations and soliton solutions are obtained. Solitonic propagation and elastic collisions between/among the two- and three-solitons are discussed analytically and graphically. It can be seen that, after each collision, solitonic shapes and amplitudes keep invariant except for some phase shifts, and the smaller-amplitude soliton moves faster and overtakes the larger.

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Notes

  1. This operator is defined as \(R=D^{2}+4U+2U_{x}D^{-1}\) with \(D\) denoting the total derivative with respect to \(x\), \(D^{-1}u=\int u dx\), and \(U\) being a function of \(x\) and \(t\) [34–38].

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Acknowledgments

Thank you very much for editors and referees’ kind letter with valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.

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Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Yun-Po Wang, Bo Tian, Ming Wang, Yu-Feng Wang, Ya Sun & Xi-Yang Xie

  2. State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China

    Bo Tian

  3. Key Laboratory of Information Photonics and Optical Communications, Ministry of Education, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Bo Tian

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  1. Yun-Po Wang
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Correspondence to Yun-Po Wang.

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Wang, YP., Tian, B., Wang, M. et al. Bäcklund transformations and soliton solutions for a (2 + 1)-dimensional Korteweg–de Vries-type equation in water waves. Nonlinear Dyn 81, 1815–1821 (2015). https://doi.org/10.1007/s11071-015-2109-y

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