Qualitative analysis for a new integrable two-component Camassa-Holm system with peakon and weak kink solutions
In this paper, authors consider the following integrable two-component Camassa-Holm system with both quadratic and cubic nonlinearity problem: \[ \begin{cases} &m_t+\frac12[(uv-u_xv_x)m]_x-\frac12[(uv_x-u_xv)m]+bu_x=0,\\ &n_t+\frac12[(uv-u_xv_x)n]_x+\frac12[(uv_x-u_xv)n]+bv_x=0, \end{cases} \] where \(m = u -u_{xx}\), \(n = v -v_{xx}\) , and \(b\) takes an arbitrary value. It is the first integrable system that admits weak kink and kink-peakon interactional solutions. It can be reduced to the CH equation, the cubic CH equation. They 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.
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