Weak well-posedness for a modified two-component Camassa–Holm system

In this paper, we consider the weak well-posedness for a modified two-component Camassa–Holm system with initial data z0=(u0,γ0)∈(H1(R)∩W1,∞(R))×(H1(R)∩W1,∞(R)). At first, by introducing a new set of independent variables, we transform the modified two-component Camassa–Holm system into an ODE system. Then, we prove the local existence and uniqueness of the solution to the ODE system and obtain the unique solution to the original system. Finally, we investigate the weak continuous dependence on initial conditions to the modified two-component Camassa–Holm system.