linear superpositions of single peakons with time-dependent amplitudes and speeds. The CH equation also possesses a large class of solutions in which smooth initial data evolves to form a cusped wave in a finite time (i.e., a gradient blow up) while the wave amplitude remains bounded. These features are shared by many other equations, all of which belong to a general family of peakon equations [@AncRec] $u_t -u_{txx} + f(u,u_x)(u-u_{xx}) + (g(u,u_x)(u-u_{xx}))_x=0$ where $f(u,u_x)$ and $g(u,u_x)$ are arbitrary non-singular functions.

The CH equation is also an integrable system [@CamHol; @FisSch; @FucFok; @Qia03]. In particular, it has a Lax pair, a bi-Hamiltonian formulation, and an infinite hierarchy of symmetries and conservation laws generated by a recursion operator. Moreover, the CH equation is related to the integrable hierarchy that contains the Korteweg-de Vries (KdV) equation $v_t + vv_x +v_{xxx}=0$, which itself is an integrable system arising from the theory of shallow water waves. Firstly, the CH equation can be obtained from a negative flow in the KdV hierarchy by a hodograph transformation [@Fuc]. Secondly, the CH equation also can be derived as a potential flow by applying the KdV recursion operator to the flow generated by $u_x$ where $v=u-u_{xx}$ relates the variables 