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Quasi-periodic wave solutions and two-wave solutions of the KdV–Sawada–Kotera–Ramani equation

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Abstract

The two-wave solutions of the KdV–Sawada–Kotera–Ramani equation are studied in this paper. By reducing this high-order wave equation into two associated solvable ordinary differential equations, we derive the two-wave solutions in the form \(u(x,t)=U(x-c_1t)+V(x-c_2t)\) which includes solitary wave solutions, periodic solutions and quasi-periodic wave solutions by letting \(c_1=c_2\). We obtain a family of new exact two-wave solutions combined by a solitary wave and a periodic wave with two different wave speeds. These new exact two-wave solutions are neither periodic nor quasi-periodic wave solutions but approximating periodic wave solutions as time tends to infinity. The process of translation of the two-wave solution combined by two solitary wave solutions is illustrated by simulation. The approach presented in this work might be applied to study the bifurcation of multi-wave solutions of some important high-order nonlinear wave model equations.

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Acknowledgments

L. Zhang thanks the North-West University for the postdoctoral fellowship. This work is supported by the National Nature Science Foundation of China (Nos. 11672270, 11501511). This research is also supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LY15A010021. The first author is also partially supported by DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS). We would like to thank anonymous referees for their valuable comments which highly improve the presentation of this work.

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Authors and Affiliations

  1. Department of Mathematics, School of Science, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, 310018, China

    Lijun Zhang

  2. International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa

    Lijun Zhang & Chaudry Masood Khalique

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  1. Lijun Zhang
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  2. Chaudry Masood Khalique
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Correspondence to Lijun Zhang.

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Zhang, L., Khalique, C.M. Quasi-periodic wave solutions and two-wave solutions of the KdV–Sawada–Kotera–Ramani equation. Nonlinear Dyn 87, 1985–1993 (2017). https://doi.org/10.1007/s11071-016-3168-4

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  • DOI: https://doi.org/10.1007/s11071-016-3168-4

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