Painlevé II asymptotics near the leading edge of the oscillatory zone for the Korteweg—de Vries equation in the small‐dispersion limit

Abstract

In the small dispersion limit, solutions to the Korteweg-de Vries equation develop an interval of fast oscillations after a certain time. We obtain a universal asymptotic expansion for the Korteweg-de Vries solution near the leading edge of the oscillatory zone up to second order corrections. This expansion involves the Hastings-McLeod solution of the Painlevé II equation. We prove our results using the Riemann-Hilbert approach.

On the Increasing Tritronquée Solutions of the Painlevé-II Equation

“…For instance, similarity solutions of the modified Korteweg-de Vries equation satisfy (1.1) [16]. The oscillations appearing near the leading edge of the dispersive shock wave generated from a wide class of initial data in the weakly-dispersive Korteweg-de Vries equation have a universal profile corresponding to the Hastings-McLeod solution of (1.1) with α = 0 [12]. The real graphs of the rational solutions of (1.1) for integer values of α determine the locations of kinks in spacetime near a point where generic initial data for the semiclassical sine-Gordon equation crosses the separatrix in the phase portrait of the simple pendulum in a transversal manner [9].…”

The KdV Hierarchy: Universality and a Painlevé Transcendent

“…Differentiating the integral equation (3.84) and (4.3) in x, one similarly obtains the estimates 4) using the fact that v R,x L 2 (Σ R ) = O(ǫ). Now by (3.76), we have that S(λ) = R(λ)P (∞) (λ) for large λ, which implies by (3.35) that…”

“…To get a feeling for the small ǫ behavior of the jump matrix v T , we need to have information about the reflection coefficient r 0 (λ; ǫ) for small values of ǫ. We have the following results, see [3,4] in combination with [37], for any choice of δ 1 > 0.…”

“…[8,9,7,11], and it is well-known that this problem can be solved explicitly in terms of the Airy function. The unique solution to this precise problem has been constructed in [4]. In the singular case later on, we will use the same model RH problem, but with a different value for θ.…”

Whitham Modulation Equations and Application to Small Dispersion Asymptotics and Long Time Asymptotics of Nonlinear Dispersive Equations

“…It follows that the wave number and the frequency depends weakly on time and too. We are going to derive the equations of A = A(x,t), B = B(x,t) and V = V (x,t) in such a way that (10) is an approximate solution of the KdV equation (5) up to subleading corrections. We are going to apply the nonlinear analogue of the WKB theory introduced in [19].…”