Preliminary group classification for the nonlinear wave equation utt=f(x,u)uxx+g(x,u)utt=f(x,u)uxx+g(x,u)

A classification is given for the nonlinear wave equation utt=f(x,u)uxx+g(x,u)utt=f(x,u)uxx+g(x,u) admitting an extension by one dimension of the principal Lie algebra of the equation under consideration. The paper is one of the few applications of an algebraic approach to the problem of group classification: the method of preliminary group classification. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. The result of the work is a wide class of equations summarized in table form.