Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa–Holm equation

The dynamical behavior of travelling wave solutions in the Generalized Camassa–Holm equation ut + 2kux − uxxt + aumux = 2uxuxx + uuxxx is analyzed by using the bifurcation theory and the method of phase portraits analysis. The condition under which compactons and cusp waves appear are also given. In addition, the reason for solitary cusp wave and periodic cusp wave to exist is highlighted.