A Riemann-Hilbert approach to the two-component modified Camassa-Holm system with step-like boundary conditions

We develop a Riemann--Hilbert formalism for the Cauchy problem for the two-component modified Camassa--Holm (2-mCH) system on the real line under non-zero step-like boundary conditions. Starting from the corresponding Lax pair, we carry out the direct spectral analysis and establish the analytic, symmetry, and asymptotic properties of the eigenfunctions and spectral data determined by the initial conditions. These results are then used to formulate a matrix Riemann--Hilbert problem whose solution provides a reconstruction formula for the solution of the original Cauchy problem.