An enormous diversity of soliton solutions to the (2+1)-dimensional extended shallow water wave equation using three analytical methods

Abstract

In this paper, we obtain a variety of analytical wave solutions of a ([Formula: see text])-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the [Formula: see text]-expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering.