Qualitative Analysis for a New Integrable Two-Component Camassa-Holm System with Peakon and Weak Kink Solutions

This paper is devoted to a new integrable two-component Camassa-Holm system with peaked solitons (peakons) and weak-kink solutions. It is the first integrable system that admits weak kink and kink-peakon interactional solutions. In addition, the new system includes both standard (quadratic) and cubic Camassa-Holm equations as two special cases. In the paper, we first establish the local well-posedness for the Cauchy problem of the system, and then derive a precise blow-up scenario and a new blow-up result for strong solutions to the system with both quadratic and cubic nonlinearity. Furthermore, its peakon and weak kink solutions are discussed as well.