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

Abstract

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