Lie symmetry analysis and new periodic solitary wave solutions of (3+1)-dimensional generalized shallow water wave equation

TL;DR

This paper uses Lie symmetry analysis to derive new exact periodic solitary wave solutions for the (3+1)-dimensional generalized shallow water wave equation, revealing complex wave structures relevant to physical systems.

Contribution

The paper introduces a novel application of Lie symmetry methods to find explicit solutions of the (3+1)-dimensional GSWW equation, including new wave types and reductions to known equations.

Findings

Derived new periodic solitary wave solutions

Obtained cross-kink soliton and breather-type solutions

Reduced the equation to various ODEs and the (2+1)-D BLMP equation

Abstract

Many important physical situations such as fluid flows, marine environment, solid-state physics and plasma physics have been represented by shallow water wave equation. In this article, we construct new solitary wave solutions for the (3+1)-dimensional generalized shallow water wave (GSWW) equation by using Lie symmetry method. A variety of analytic (closed-form) solutions such as new periodic solitary wave, cross-kink soliton and doubly periodic breather-type solutions have been obtained by using invariance of the concerned (3+1)-dimensional GSWW equation under one-parameter Lie group of transformations. Lie symmetry transformations have applied to generate the different forms of invariant solutions of the (3+1)-dimensional GSWW equation. For different Lie algebra, Lie symmetry method reduces (3+1)-dimensional GSWW equation into various ordinary differential equations (ODEs) while one of the Lie algebra, it is transformed into the well known (2+1)-dimensional BLMP equation. It is affirmed that the proposed techniques are convenient, genuine and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Under the suitable choices of arbitrary functions and parameters, 2D, 3D and contour graphics to the obtained results of GSWW equation are also analyzed graphically.