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Kink-like wave and compacton-like wave solutions for generalized KdV equation

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Abstract

By using the bifurcation method of dynamical systems and numerical simulation approach of differential equations, we investigate generalized KdV equation \(u_t=u^{2}u_{x}-u^{2}u_{xxx}-4uu_xu_{xx}-(u_x)^3\). Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, some interesting phenomena are revealed, that is, under some conditions, the periodic wave can become the kink-like wave and compacton-like wave, respectively, and the solitary wave can become the kink-like wave. The exact kink-like wave and compacton-like wave solutions with implicit or parameter expressions are given.

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Acknowledgments

All authors wish to thank the editor and the anonymous referee for many valuable suggestions leading to an improvement of this paper. This work is supported by the Science Foundation of Shaoguan University (201320501), National Natural Science Foundation of China (11401448).

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Authors and Affiliations

  1. School of Mathematics and Information Sciences, Shaoguan University, Shaoguan, 512005, People’s Republic of China

    Shaoyong Li

  2. Department of Mathematics, South China University of Technology, Guangzhou, 510640, Guangdong, People’s Republic of China

    Shaoyong Li & Zhengrong Liu

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  1. Shaoyong Li
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  2. Zhengrong Liu
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Correspondence to Shaoyong Li.

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Li, S., Liu, Z. Kink-like wave and compacton-like wave solutions for generalized KdV equation. Nonlinear Dyn 79, 903–918 (2015). https://doi.org/10.1007/s11071-014-1710-9

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  • DOI: https://doi.org/10.1007/s11071-014-1710-9

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