Uniqueness of global conservative weak solutions for the modified two-component Camassa–Holm system

Abstract

In this paper, we consider the uniqueness of global-in-time conservative weak solutions for the modified two-component Camassa–Holm system on real line. The strategy of proof is based on characteristics. Given a conservative weak solution, an equation is introduced to single out a unique characteristic curve through each initial point coordinate transformation into the Lagrangian coordinates. We prove that the Cauchy problem of the modified two-component Camassa–Holm system with initial data $$z_0=(u_0,\gamma _0)\in H^1(\mathbb {R})\times (H^1(\mathbb {R})\cap W^{1,\infty }(\mathbb {R}))$$ has a unique global conservative weak solution.