Summary: We study the peakons and the periodic cusp wave solutions of the equation
with , which we will call the generalized Camassa-Holm equation, or simply the GCH equation, for it is the CH Camassa-Holm equation. Camassa and Holm showed that the CH equation has a class of new solitary wave solutions called “peakons”.
Using the bifurcation method of the phase plane, we first construct peakons and show that is the peakon bifurcation parameter value for the GCH equation. Then we construct some smooth periodic wave solutions, periodic cusp wave solutions, and oscillatory solitary wave solutions, and show their convergence when either the parameter or the wave speed varies. We also illustrate how to identify the existence of peakons and periodic cusp waves from the phase portraits. It seems that the GCH equation is a good example to understand the relationships among peakons, periodic cusp waves, oscillatory solitary waves and smooth periodic wave solutions.