Abstract: Firstly, the explicit new travelling wave and group invariant solutions of the Liouville equation were given by using the methods of extended tanh-function and Lie group analysis, respectively. Secondly, the ψ(ξ) expansion method for solving nonlinear partial differential equations was constructed by using the reduced transformation equation of the Liouville equation and its exact solutions. Finally, some explicit new travelling wave solutions of the Kawahara equation and (3+1)-dimensional Kadomtsev-Petviashvili equation were given by using ψ(ξ) expansion method.

Key words: Liouville equation； ψ(ξ) expansion method； exact solution, travelling wave solution