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 modulated finite-gap elliptic function and a finite number of solitons.