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Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation, bilinear representation, and Wronskian solution

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Abstract

With symbolic computation, this paper investigates some integrable properties of a two-dimensional generalization of the Korteweg-de Vries equation, i.e., the Bogoyavlensky–Konoplechenko model, which can govern the interaction of a Riemann wave propagating along the \(y\)-axis and a long wave propagating along the \(x\)-axis. Within the framework of Bell-polynomial manipulations, Bell-polynomial expressions are firstly given, which then are cast into bilinear forms. The \(N\)-soliton solutions in the form of an \(N\)th-order polynomial in the \(N\) exponentials and in terms of the Wronskian determinant are, respectively, constructed with the Hirota bilinear method and Wronskian technique. Bilinear Bäcklund transformation is also derived with the achievement of a family of explicit solutions.

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Acknowledgments

We would like to express our sincere thanks to the referees and the editor Edita Alfred for their valuable suggestions. This work is supported by the National Natural Science Foundation of China under Grant No. 61308018, by China Postdoctoral Science Foundation under Grant No. 2012M520154, by the Fundamental Research Funds for the Central Universities of China (2013JBM088), and partially by the Project of State Key Laboratory of Rail Traffic Control and Safety (No. RCS2012ZT004), Beijing Jiao Tong University. JL is supported by National Natural Science Foundation of China under Grant No. 11101421, and the Special Foundation for Young Scientists of Institute of Remote Sensing and Digital Earth of Chinese Academy of Sciences (No. Y1S01500CX)

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiao Tong University, Beijing, 100044, People’s Republic of China

    Xing Lü

  2. State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiao Tong University, Beijing,  100044, People’s Republic of China

    Xing Lü

  3. State Key Laboratory of Remote Sensing Science, Jointly Sponsored by the Institute of Remote Sensing and Digital Earth of Chinese Academy of Sciences and Beijing Normal University, Beijing,  100101, People’s Republic of China

    Juan Li

  4. Demonstration Centre, Spaceborne Remote Sensing National Space Administration, Beijing,  100101, People’s Republic of China

    Juan Li

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Lü, X., Li, J. Integrability with symbolic computation on the Bogoyavlensky–Konoplechenko model: Bell-polynomial manipulation, bilinear representation, and Wronskian solution. Nonlinear Dyn 77, 135–143 (2014). https://doi.org/10.1007/s11071-014-1279-3

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