Contact transformation group classification of nonlinear wave equations (Q1577389)

This paper studies the class of PDE's of the form \(u_{tt}=f(x,u_x)u_{xx}+g(x,u_x)\) for arbitrary functions \(f(x,u_x)\) and \(g(x,u_x)\). The authors reduce the problem of classifying these equations with respect to admitted contact transformation groups to the study of point symmetries of the equivalent system of first-order quasilinear equations \(v_t=a(x,v)w_x\) and \(w_t=b(x,v)v_x\). For this system, the principal Lie algebra admitted by the whole class is two-dimensional. Conditions are given for \(a\) and \(b\) that tell when it is possible to extend the principal Lie algebra of the system.