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The \(\varvec{(2+1)}\)-dimensional Konopelchenko–Dubrovsky equation: nonlocal symmetries and interaction solutions

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Abstract

The nonlocal symmetries for the \((2+1)\)-dimensional Konopelchenko–Dubrovsky equation are obtained with the truncated Painlevé method and the Möbious (conformal) invariant form. The nonlocal symmetries are localized to the Lie point symmetries by introducing auxiliary dependent variables. The finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. The multi-solitary wave solution is presented with the finite symmetry transformations of a trivial solution. In the meanwhile, symmetry reductions in the enlarged systems are studied by the Lie point symmetry approach. Many explicit interaction solutions between solitons and cnoidal periodic waves are discussed both in analytical and in graphical ways.

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Acknowledgments

This work was supported by Zhejiang Provincial Natural Science Foundation of China under Grant (Nos. LZ15A050001 and LQ16A010003) and the National Natural Science Foundation of China under Grant (Nos. 11305106 and 11505154)

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Authors and Affiliations

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

    Bo Ren

  2. Department of Physics, Zhejiang Ocean University, Zhoushan, 316000, China

    Xue-Ping Cheng

  3. Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang Province, Zhoushan, 316022, China

    Xue-Ping Cheng

  4. Institute of Nonlinear Physics, Zhejiang Normal University, Jinhua, 321004, China

    Ji Lin

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  1. Bo Ren
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  2. Xue-Ping Cheng
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  3. Ji Lin
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Correspondence to Bo Ren.

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Ren, B., Cheng, XP. & Lin, J. The \(\varvec{(2+1)}\)-dimensional Konopelchenko–Dubrovsky equation: nonlocal symmetries and interaction solutions. Nonlinear Dyn 86, 1855–1862 (2016). https://doi.org/10.1007/s11071-016-2998-4

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  • DOI: https://doi.org/10.1007/s11071-016-2998-4

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