Well-posedness of a class of solutions to an integrable two-component Camassa–Holm system

Abstract

In this paper, we study a class of solutions to the Cauchy problem for an integrable two-component Camassa–Holm system with the initial data ( u 0 , ρ 0 − 1 ) ∈ ( H 1 ( R ) ∩ W 1 , ∞ ( R ) ) × ( L 2 ( R ) ∩ L ∞ ( R ) ) . Based on characteristics, we study a corresponding ODE and obtain a unique local solution by applying the contraction mapping principle. We prove local existence and uniqueness of the solution to the Camassa–Holm system by constructing a solution obtained from the ODE and studying the regularity of the solution. Finally, we show continuous dependence of the solution on the initial data in some weak sense.