Riemann–Hilbert problem and soliton resolution for the two‐component Camassa–Holm system

Abstract In this paper, we analyze the Cauchy problem for the two‐component Camassa–Holm (2CH) system in weighted Sobolev spaces, achieving two key results. First, we establish a new irregular Riemann–Hilbert (RH) problem, without the normalization conditions, for the associated Cauchy problem. Second, we identify six main asymptotic sectors in the space–time region and derive the leading‐order asymptotics for these sectors using the ‐nonlinear steepest descent method. This confirms the soliton resolution conjecture for the 2CH system. Notably, the 2CH system exhibits unique characteristics within the RH formalism, distinguishing it from the classical Camassa–Holm (CH) equation. We prove that the reconstruction process remains unique even when the RH problem lacks uniqueness. Additionally, numerical simulations are provided to validate these asymptotic results.