An integrable system with peakon, complex peakon, weak kink, and kink-peakon interactional solutions

In this paper, we study an integrable system with both quadratic and cubic nonlinearity: mt=bux+12k1[m(u2−ux2)]x+12k2(2mux+mxu),m=u−uxx, where b, k1 and k2 are arbitrary constants. This model is kind of a cubic generalization of the Camassa-Holm (CH) equation: mt+mxu+2mux=0. The equation is shown integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. In the case b=0, peaked soliton (peakon), complex peakon, and multi-peakon solutions are studied. In particular, the two-peakon dynamical system is explicitly presented and their collisions are investigated in details. In the case b ≠ 0, the weak kink and kink-peakon interactional solutions are found for the first time. Significant difference from the CH equation is analyzed through a comparison. In the paper, we also investigate all possible smooth one-soliton solutions for the system.