Sul limite di piccola dispersione per le equazioni di Camassa-Holm e Whitham

In this talk we consider the so-called small dispersion limit of solutions of the Cauchy problem of the Camassa Holm (CH) equation for smooth initial data such that wave-breaking does not occur. For initial data in this class, the numerical solution of (CH) develops a zone of fast oscillations, as for the small dispersion limit of the Korteweg-de Vries equation. An asymptotic description of these oscillations is presented together with a conjecture for the phase of the asymptotic solution. The description proposed is in agreement with both numerical evidence and a recent conjecture by Dubrovin.