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

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