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Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation

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Abstract

In the present paper, a general solution involving three arbitrary functions for the generalized (2+1)-dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equation, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multisymplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)-dimensional KdV-mKdV equation based on the multisymplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from the general solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent to the Preissmann scheme. From the results of the numerical experiments, we can conclude that the multi-symplectic schemes can accurately simulate the periodic wave solutions of the generalized (2+1)-dimensional KdV-mKdV equation while preserve approximately the conservation laws.

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Authors and Affiliations

  1. School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnincal University, 710072, Xi’an, China

    Wei-Peng Hu, Zi-Chen Deng, Yu-Yue Qin & Wen-Rong Zhang

  2. State Key Laboratory of Mechanical System & Vibration, Shanghai Jiao Tong University, 200240, Shanghai, China

    Wei-Peng Hu

  3. State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, 116023, Dalian, China

    Zi-Chen Deng

Authors
  1. Wei-Peng Hu
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  2. Zi-Chen Deng
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  3. Yu-Yue Qin
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  4. Wen-Rong Zhang
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Correspondence to Wei-Peng Hu.

Additional information

The project was supported by the National Natural Science Foundation of China (11002115, 10972182, 11172239), the Science Foundation of Aviation of China (2010ZB53021), the China Postdoctoral Science Special Foundation (201003682), 111 project (B07050) to the Northwestern Polytechnical University, the NPU Foundation for Fundamental Research (JC200938, JC20110259), the Doctoral Program Foundation of Education Ministry of China (20106102110019), the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21) and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802).

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Hu, WP., Deng, ZC., Qin, YY. et al. Multi-symplectic method for the generalized (2+1)-dimensional KdV-mKdV equation. Acta Mech Sin 28, 793–800 (2012). https://doi.org/10.1007/s10409-012-0070-2

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  • DOI: https://doi.org/10.1007/s10409-012-0070-2

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