Group invariant solutions and Conservation laws of the nonlinear Gardner-Kawahara equation

TL;DR

This paper applies Lie symmetry analysis to the nonlinear Gardner-Kawahara equation to find invariant solutions, conservation laws, and their physical implications, supported by explicit solutions and numerical simulations.

Contribution

It provides a comprehensive analysis of the Gardner-Kawahara equation, including exact solutions, conservation laws, and physical interpretations using symmetry methods.

Findings

Exact explicit solutions of the Gardner-Kawahara equation obtained.

Complete set of local conservation laws constructed.

Numerical simulations illustrate physical relevance of solutions.

Abstract

The present article studies the potential form of the nonlinear Gardner-Kawahara equation through the perspective of Lie symmetry analysis. Lie symmetry analysis was used to investigate abundant group-invariant solutions of the nonlinear Gardner-Kawahara equation. This method is used to provide geometric vector fields, as well as their commutative and adjoint relations. In this article, we have obtained the exact solution of the nonlinear Gardner-Kawahara equation in explicit form by different significant methods. Numerical simulation is used to explain the physical relevance of invariant solutions in 3D and 2D graphs. Finally, by the conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation. The conserved currents discovered in this study can help us better comprehend some of the physical processes that the underlying equations predict.