Optimal System, Reductions, and Conservation Laws of a Nonlinear Damped Klein-Gordon- Fock Equation

A detailed Lie symmetry analysis of the nonlinear damped Klein-Gordon Fock equation: $u_{tt}\,+\alpha (u)\,u_{t}\,=\,u_{xx}\,+\,f(u)$ is addressed in this paper. Applying the Lie symmetry method, a comprehensive Lie group classification is performed for the arbitrary smooth functions $\alpha (u)$ and $f(u)$ present in the equation, leading to two distinct cases. Additionally, for each case an optimal system of one-dimensional subalgebras is derived, which is a minimal set of all the linearly independent symmetry generators without redundant symmetries. Using the similarity transformation method, the above-mentioned partial differential equation is reduced into a set of ordinary differential equations. In certain cases, several exact invariant solutions encompassing the travelling wave solutions and soliton waves are obtained. The graphs of the soliton solutions and traveling wave solutions are also presented. Finally, the conservation laws are identified via the partial Noether approach, leading to distinct cases with several subcases. The derived conservation laws provide valuable tools for examining the dynamics and stability of physical systems, making this research suitable to a range of scientific studies.