The bifurcation and peakons for the special C (3, 2, 2) equation

In this paper, we investigate a special C(3, 2, 2) equation $$\begin{array}{@{}rcl@{}} u_{t}+ku_{x}-u_{xxt}+3(u^{3})_{x}=u_{x}(u^{2})_{xx}+u(u^{2})_{xxx}. \end{array} $$ The bifurcation and some new exact representations of peakons, bell-shaped solitary wave solutions and periodic cusp wave solutions for the equation are obtained using the qualitative theory of dynamical systems. It is shown that the peakons are actually the limit of bell-shaped solitary waves and periodic cusp waves. Moreover, a new characteristic of non-smooth solutions, two peakons coexisting for the same wave speed, is found. Some previous results are extended.