Wave patterns associated with a singular line for a bi-Hamiltonian system

The evolution of wave patterns of a generalized KdV equation is investigated in this Letter. For the topological vector filed, a singular line can be observed which separates the phase trajectories on the phase plane into two parts. All possible waves including compactons, solitary waves, smooth periodic waves and non-smooth periodic waves with peaks as well as the existence conditions have been presented. Specially, a new type of waves called as kink-compactons which combine both half structures of compacton and kink-wave can be found.