  The Cauchy problem for the Korteweg de Vries (KdV) equation with small
dispersion of order $\e^2$, $\e\ll 1$, is characterized by the appearance of a
zone of rapid modulated oscillations. These oscillations are approximately
described by the elliptic solution of KdV where the amplitude, wave-number and
frequency are not constant but evolve according to the Whitham equations.
Whereas the difference between the KdV and the asymptotic solution decreases as
$\epsilon$ in the interior of the Whitham oscillatory zone, it is known to be
only of order $\epsilon^{1/3}$ near the leading edge of this zone. To obtain a
more accurate description near the leading edge of the oscillatory zone we
present a multiscale expansion of the solution of KdV in terms of the
Hastings-McLeod solution of the Painlev\'e-II equation. We show numerically
that the resulting multiscale solution approximates the KdV solution, in the
small dispersion limit, to the order $\epsilon^{2/3}$.
