Riemann-Hilbert approach to the modified Camassa-Holm positive flow with step-like initial data

We study the Cauchy problem for the modified Camassa-Holm positive flow with step-like initial data. Using the Riemann-Hilbert approach, we first establish the basic Riemann-Hilbert problem, which involves three distinct forms of jump matrices defined on corresponding intervals of the real axis. The reconstruction formula for the solution is then derived by analyzing the behavior of the eigenfunctions at λ = 0 and λ = ∞. Based on the nonlinear steepest descent method, we deform the basic Riemann-Hilbert problem and isolate the parts that make the dominant contribution to the solution through rigorous error estimation. Finally, the long-time asymptotic behavior of the solution is obtained.