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A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves

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Abstract

A nonlocal two-layer fluid model is constructed through a simple symmetry reduction from the local one. Then, a general variable coefficients nonlocal KdV (VCKdV) equation with shifted space parity and delayed time reversal is derived from it by using multi-scale expansion method, with and without the so-called \(y-\)average method, respectively. A non-auto-Bäcklund transformation between the VCKdV equation and a constant coefficients KdV (CCKdV) equation is established. By using this transformation, various exact solutions of the VCKdV equation can be obtained from the seed solutions of the CCKdV equation. As some concrete examples, one solitary wave solution and two kinds of periodic wave solutions are given. Due to the inclusion of arbitrary functions in these solutions, they possess abundant dynamical behaviors with some of them analyzed graphically.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant No. 11405110 and the Natural Science Foundation of Zhejiang Province of China under Grant No. LY18A050001.

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  1. Institute of Nonlinear Science, Shaoxing University, Shaoxing, 312000, China

    Xi-zhong Liu

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  1. Xi-zhong Liu
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Correspondence to Xi-zhong Liu.

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The authors declare that they have no conflicts of interest to this work. There is no professional or other personal interest of any nature or kind in any product that could be construed as influencing the position presented in the manuscript entitled.

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Liu, Xz. A nonlocal variable coefficient KdV equation: Bäcklund transformation and nonlinear waves. Eur. Phys. J. Plus 135, 113 (2020). https://doi.org/10.1140/epjp/s13360-020-00178-1

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  • DOI: https://doi.org/10.1140/epjp/s13360-020-00178-1

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