Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized degasperis–procesi equation

In this paper, we employ the bifurcation theory of planar dynamical systems to study the smooth and nonsmooth traveling wave solutions of the generalized Degasperis-Procesi equationut-uxxt+4umux=3uxuxx+uuxxx.ut-uxxt+4umux=3uxuxx+uuxxx.The parameter condition under which peakons, compactons and periodic cusp wave solutions exist is given. The numerical simulation results show the consistence with the theoretical analysis at the same time.