Abstract: The travelling wave solutions (TWS) in a class of P.D.E. is studied. The travelling wave equation of this P.D.E. is a planar cubic polynomial system in three-parameter space. The study for TWS became the topological classifications of bifurcations of phase portraits defined by the planar system. By using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(ξ), and considering discontinuity of the right side of the equation of TWS when ξ=x-ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.

Key words: nonlinear wave equation, solitary travelling wave, periodic travelling wave, dissmoothness of wave