Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations

Abstract

In this paper, the combination of Lie symmetry analysis and the dynamical system method is performed on the mixed second-order sine-Gordon equation, all of the geometric vector fields of the sine-Gordon equation, the generalized nonlinear wave equation and its special case, the Liouville equation are presented. Then, the symmetry reductions and exact solutions to such nonlinear wave equations are considered. Especially, the bifurcations of the sine-Gordon equation are obtained, and the exact explicit traveling wave solutions are investigated by the dynamical system method. To guarantee the existence of the traveling wave solutions, all of the parameter conditions are determined.