Numerical Study of a Multiscale Expansion of the Korteweg De Vries Equation and Painlevé-ii Equation

The Cauchy problem for the Korteweg de Vries (KdV) equation with small dispersion of order ǫ, ǫ ≪ 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 ǫ in the interior of the Whitham oscillatory zone, it is known to be only of order ǫ1/3 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 Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order ǫ2/3.