Riemann–Hilbert problem for Camassa–Holm equation with step-like initial data

We consider the Cauchy problem for the Camassa–Holm equation. The initial data is supposed to be a step-like function, i.e. it tends to different limits as spatial variable tends to plus or minus infinity. We derive a parametric representation of the solution of the initial value problem in terms of the solution of an associated Riemann–Hilbert problem; this allows to study effectively the long-time asymptotic behavior of the solution by using the nonlinear steepest descent method, at least for two sectors of the (x,t)(x,t)-half-plane.