Asymptotics of step-like solutions for the Camassa-Holm equation

We study the long-time asymptotics of solution of the Cauchy problem for the Camassa-Holm equation with a step-like initial datum. By using the nonlinear steepest descent method and the so-called $g$-function approach, we show that the Camassa-Holm equation exhibits a rich structure of sharply separated regions in the $x,t$-half-plane with qualitatively different asymptotics, which can be described in terms of a sum of modulated finite-gap hyperelliptic or elliptic functions and a finite number of solitons.