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Residual symmetry, Bäcklund transformation and CRE solvability of a (\(\mathbf{2}{\varvec{+}}{} \mathbf{1}\))-dimensional nonlinear system

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Abstract

In this paper, the truncated Painlevé expansion is employed to derive a Bäcklund transformation of a (\(2+1\))-dimensional nonlinear system. This system can be considered as a generalization of the sine-Gordon equation to \(2+1\) dimensions. The residual symmetry is presented, which can be localized to the Lie point symmetry by introducing a prolonged system. The multiple residual symmetries and the nth Bäcklund transformation in terms of determinant are obtained. Based on the Bäcklund transformation from the truncated Painlevé expansion, lump and lump-type solutions of this system are constructed. Lump wave can be regarded as one kind of rogue wave. It is proved that this system is integrable in the sense of the consistent Riccati expansion (CRE) method. The solitary wave and soliton–cnoidal wave solutions are explicitly given by means of the Bäcklund transformation derived from the CRE method. The dynamical characteristics of lump solutions, lump-type solutions and soliton–cnoidal wave solutions are discussed through the graphical analysis.

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Acknowledgements

This work is supported by the National Natural Science Foundation of China under Grant No. 41474102.

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Zhonglong Zhao & Bo Han

  2. Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada

    Zhonglong Zhao

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  1. Zhonglong Zhao
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Correspondence to Zhonglong Zhao.

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Zhao, Z., Han, B. Residual symmetry, Bäcklund transformation and CRE solvability of a (\(\mathbf{2}{\varvec{+}}{} \mathbf{1}\))-dimensional nonlinear system. Nonlinear Dyn 94, 461–474 (2018). https://doi.org/10.1007/s11071-018-4371-2

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  • DOI: https://doi.org/10.1007/s11071-018-4371-2

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