Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

Abstract

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity utt+α(u)ut=(uβux)x+f(u). We carried out a complete Lie symmetry classification by finding forms for α(u) and f(u). This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach.

Keywords

• non-linear damped Klein–Gordon equation
• Lie symmetries
• optimal systems
• reductions
• invariant solutions
• conservation laws