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

Abstract

In this paper, the nonlinear Gardner-Kawahara (NLGK) equation has been investigated by using the Lie symmetry method. The symmetry method is applied to derive the infinitesimal generators and vector fields. The Lie symmetry method has reduced the nonlinear Gardner-Kawahara equation into a nonlinear ordinary differential equation form. In this article, we have obtained the exact solution of the NLGK equation in the explicit form by different significant methods. Further, the multiplier method has been used to derive the conservation laws of the nonlinear Gardner-Kawahara equation.

A new numerical approach to Gardner Kawahara equation in magneto‐acoustic waves in plasma physics

“…In terms of physical interpretation, the Gardner-Kawahara equation explains the solitary wave propagation in media. 1 In the study of interfacial waves between two immiscible fluids, there are such situation when the double critical conditions can occur, that is, when both the coefficients of quadratic nonlinearity and third-order dispersion vanish simultaneously. In the near-critical situation, the basic governing equation is the Gardner-Kawahara equation.…”

“…Among others, Hussein and Taha 2 have efficiently and successfully used (G'/G)-expansion method to find new solitary wave solutions of the Gardner-Kawahara equation. Swain et al 1 have investigated the equation using the Lie symmetry method. First they have reduced the equation into a nonlinear ordinary differential equation form and then obtained the exact solution in the explicit form by different significant methods.…”

A New Numerical Approach to Gardner Kawahara Equation in Magneto-Acoustic Waves in Plasma Physics

“…The equation can be widely encountered in shallow water waves having surface tension as well as in plasma physics. In terms of physical interpretation, the Gardner-Kawahara equation expalins the solitary wave propagation in media [9].In the study of interfacial waves between two immiscible fluids, there are such situation when the double critical conditions can occur, i.e. when both the coefficients of quadratic nonlinearity and third-order dispersion vanish simultaneously.…”

“…Among others, Hussein and Taha [4] have efficiently and successfully used (G'/G)-expansion method to find new solitary wave solutions of the Gardner-Kawahara equation. Swain et al [9] have investigated the equation using the Lie symmetry method. First they have reduced the equation into a nonlienar ordinary differential equation form and then obtained the exact solution in the explicit form by different significant methods.…”