Smooth and non-smooth travelling wave solutions of generalized Degasperis-Procesi equation

The dynamical behavior and special exact solutions of the generalized Degasperis-Procesi equation ut + c0ux − utxx + auux = 3uxuxx + uuxxx is analyzed by using bifurcation theory and the method of phase portraits analysis. As a result, the analytic expressions of peakon solutions, compacton solutions and periodic cusp wave solutions are obtained. And the condition under which peakon and compacton solutions appear are also given. In the meantime, the reason that solitary cusp wave and periodic cusp wave appear is given.