Lie group classification and conservation laws of a (2+1)-dimensional nonlinear damped Klein–Gordon Fock equation

Abstract

In this article, the $(2+1)$-dimensional nonlinear damped Klein–Gordon Fock equation is studied using the classical Lie symmetry method. A complete Lie group classification is conducted to derive the specific forms of the arbitrary smooth functions included in the equation, resulting in two distinct cases. Using the similarity transformation method, the reductions of the considered equation in the form of ordinary differential equations are obtained. Several invariant solutions including the traveling wave solutions and soliton solutions of the $(2+1)$-dimensional nonlinear damped Klein–Gordon Fock equation are uncovered. Also, the results are represented through 2D and 3D graphs with their physical interpretations. Notably, using the partial Lagrangian method, the conservation laws are derived, which also yield two separate cases with several subcases. These results offer better insights into the solution properties of nonlinear damped Klein–Gordon Fock equation and other complex nonlinear wave equations.