Lie Symmetry Structure of Nonlinear Wave Equations in (n+1)-Dimensional Space-time

We study Lie point symmetry structure of generalized nonlinear wave equations of the form □u=F(x,u,∇u) where □ is the (n+1)-dimensional space-time wave (or d’Alembert) operator, x∈Rn+1 (n≥2). We find the equivalence groups of this class and its subclass where the first order derivatives are absent. We then determine the symmetry group as a special case of the equivalence from the invariance requirement of the nonlinearity F leading to the symmetry condition involving F. As an application we solve this condition for some specific cases of F to build physically important equations like conformally-invariant nonlinear wave and Euler–Poisson–Darboux equation. Canonical forms for allowable symmetries are also studied.