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Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach

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Abstract

Two–dimensional optimal classification is performed for two–dimensional shallow water equations with flat bottom in Cartesian coordinates. In the proof of optimality of subalgebras, adjoint actions play a very vital role. When the nonidentical adjoint actions are more complicated then the situation becomes very challenging. Here, we propose a tree structure to handle such situations. Further, the optimal set is constructed with judicious adjoint actions. Consequently group invariant solutions are obtained and graphical behavior of solutions are demonstrated for some of the inequivalent classes in optimal system. Finally, physically relevant solutions like, traveling wave solutions, namely, the kink-type and peakon-type solitons, are obtained through traveling wave transformations.

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Acknowledgements

We sincerely thank the anonymous referees for their useful comments and suggestions to improve the manuscript. Research support from, DST/ INSPIRE Fellowship/2013, Ministry of Science and Technology, Department of Science and Technology, Government of India and Sponsored Research and Industrial Consultancy, IIT Kharagpur (Ref no. IIT/SRIC/ISIRD/2013–14) are gratefully acknowledged by first and second authors, respectively.

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Authors and Affiliations

  1. Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Nagpur, India

    Purnima Satapathy

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 2, India

    T. Raja Sekhar

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  1. Purnima Satapathy
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  2. T. Raja Sekhar
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Correspondence to T. Raja Sekhar.

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Satapathy, P., Raja Sekhar, T. Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach. Eur. Phys. J. Plus 137, 1183 (2022). https://doi.org/10.1140/epjp/s13360-022-03373-4

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