The unified group classification method for the generalized nonlinear wave equation and its partial difference schemes

This paper is concerned with the generalized nonlinear second-order equation and its discrete counterpart furnished with different lattices, respectively. By the unified symmetry analysis method, the complete group classifications of the equations are performed. In the sense of point symmetry, all of the vector fields of the continuous nonlinear equation are obtained. As its special cases, the vector fields of some other important nonlinear equations are provided. Then, we develop the unified group classification method for dealing with partial difference schemes (PΔΔS), all of the point symmetries of the difference schemes are presented with respect to the given arbitrary analytic function f(u)f(u).