Group properties of generalized quasi-linear wave equations

Abstract:
Complete group classification of a class of (1+1)-dimensional generalized quasi-linear wave equations is performed by using the Lie-Ovsiannikov method and a compatibility method. Lie reductions of some truly 'variable coefficient' wave equations which are singled out from the classification results are investigated. As a result, some classes of exact solutions of these 'variable coefficient' wave equations are constructed by means of both the reductions and the additional equivalent transformations. Furthermore, The nonclassical symmetry to the generalized quasi-linear wave equation is also studied exhaustively. This enabled to obtain new exact solutions, describing the evolution of various wave patterns.