  The small dispersion limit of solutions to the Camassa-Holm (CH) equation is
characterized by the appearance of a zone of rapid modulated oscillations. An
asymptotic description of these oscillations is given, for short times, by the
one-phase solution to the CH equation, where the branch points of the
corresponding elliptic curve depend on the physical coordinates via the Whitham
equations. We present a conjecture for the phase of the asymptotic solution. A
numerical study of this limit for smooth hump-like initial data provides strong
evidence for the validity of this conjecture. We present a quantitative
numerical comparison between the CH and the asymptotic solution. The dependence
on the small dispersion parameter $\epsilon$ is studied in the interior and at
the boundaries of the Whitham zone. In the interior of the zone, the difference
between CH and asymptotic solution is of the order $\epsilon$, at the trailing
edge of the order $\sqrt{\epsilon}$ and at the leading edge of the order
$\epsilon^{1/3}$. For the latter we present a multiscale expansion which
describes the amplitude of the oscillations in terms of the Hastings-McLeod
solution of the Painlev\'e II equation. We show numerically that this
multiscale solution provides an enhanced asymptotic description near the
leading edge.
