A (2+1)-dimensional modified dispersive water-wave (MDWW) system: Lie symmetry analysis, optimal system and invariant solutions

Abstract

In this article, the authors study a (2+1)-dimensional MDWW system, which describes the non-linear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. The Lie group theoretic approach is employed to find the similarity reductions and analytic solutions of the (2+1)-dimensional MDWW system. The infinitesimal generators for the considered system are obtained under the invariance property of the Lie group of transformations. Later, we construct groups of symmetries and tables for commutation and adjoints. The adjoint table is further used to establish a one-dimensional optimal system of subalgebras. Finally, based on the optimal system, similarity reductions are obtained. A repeated process of similarity reductions reduces the governing system of partial differential equations (PDEs) into systems of ordinary differential equations (ODEs) that generate invariant solutions. Moreover, the dynamical behaviors of the obtained solutions such as multi-soliton, doubly soliton, single soliton, solitary waves, and stationary waves are graphically shown using 3D, 2D, and corresponding contour plots. Thus, physicists and mathematicians can follow complicated physical phenomena more effectively and efficiently using these graphs.