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 ϵ 2 , ϵ ≪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, wavenumber and frequency are not constant but evolve according to the Whitham equations. Whereas the difference between the KdV and the asymptotic solution decreases as ϵ in the interior of the Whitham oscillatory zone, it is known to be only of order ϵ 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 Painlevé-II equation. We show numerically that the resulting multiscale solution approximates the KdV solution, in the small dispersion limit, to the order ϵ 2/3 .