Global existence of the solution of the modified Camassa-Holm equation with step-like boundary conditions

We consider the Cauchy problem for the modified Camassa-Holm equation $$u_t+{((u^2-u_x^2)m)}_x=0,mu-u_{xx},t>0,-\mathrm{\infty }<x<+\mathrm{\infty }$$ subject to the step-like initial data: $u(x,0)\to A_1$ as $x\to -\mathrm{\infty }$ and $u(x,0)\to A_2$ as $x\to +\mathrm{\infty }$, where $0<A_1<A_2$. The goal is to establish the global existence of the solution of this problem.