Numerical study of a multiscale expansion of the Korteweg de Vries equation and Painlev\'e-II equation

TL;DR

This paper develops a multiscale expansion using Painlevé-II to improve the approximation of the KdV equation's solutions in the small dispersion limit, especially near the leading edge of oscillations.

Contribution

It introduces a novel multiscale expansion involving Painlevé-II for better accuracy near the oscillatory zone's edge in KdV solutions.

Findings

Multiscale expansion approximates KdV solutions to order ε^{2/3}.

Numerical evidence supports improved accuracy near the leading edge.

The method enhances understanding of oscillatory behavior in small dispersion regimes.

Abstract

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 $\epsilon^{2/3}$.