Symmetry Reductions, Qualitative Analysis and Conservation Law of Gardner’s Equation

Abstract

This study secures Lie symmetry analysis and traveling wave solutions to the well-known generalized form of Gardner’s equation with three dispersion sources. Using the Lie symmetry approach, infinitesimal generators and symmetry reductions of the Gardner’s equation
with triplet dispersion sources have been discovered. The equation’s reduced form is found and it is solved by the G′ G approach, which yields several solution forms. In a reduced form, bifurcation theory has been used to analyze the stability and properties of critical points
by converting the equation into an autonomous system. Phase portraits have been plotted at various critical points to depict the system’s behavior and assess its stability. The conservation laws are determined using the multipliers approach. First, the multipliers are computed based on independent variables, dependent variables and derivatives of dependent variables up to a certain order. The fluxes of the conservation laws are then obtained for conserved vector.