Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach
Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Nagpur, India
2 Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 2, India
Accepted: 6 October 2022
Published online: 27 October 2022
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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