https://doi.org/10.1140/epjp/s13360-022-03373-4
Regular Article
Analytic solutions for (2+1)-dimensional shallow water equations with flat bottom through Lie symmetry approach
1
Department of Mathematics, Visvesvaraya National Institute of Technology Nagpur, Nagpur, India
2
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 2, India
b
trajasekhar@maths.iitkgp.ac.in
Received:
4
April
2022
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.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2022. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.