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

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 has a unique global conservative weak solution.