The small dispersion limit of the Korteweg‐de Vries equation. III

Abstract

In Part I* we have shown, see Theorem 2.10, that as the coefficient of u,,, tends to zero, the solution of the initial value problem for the KdV equation tends to a limit ii in the distribution sense. We have expressed by formula (3.59), where I)? is the partial derivative with respect to x of the function $* defined in Theorem 3.9 as the solution of the variational problem formulated in (2.16), (2.17). I)* is uniquely characterized by the variational condition (3.34); its partial derivatives satisfy (3.51) and (3.52), where I is the set I' defined in The Solution Until Break-timeThe variational problem (2.16)-(2.18) contains x and t as parameters. We first investigate the case t = 0. We make the assumption, justified a posteriori, that for t < t b the set I consists of a single interval

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

“…As a drawback, the KdV solution develops oscillations with wavelength O(ǫ) for t > t c , see Figure 1. Those oscillations were already described in the classical works of Lax and Levermore [32] and investigated in more detail by Venakides [36], Tian [35], and Deift, Venakides, and Zhou [11,12]. Under mild assumptions on the initial data, u(x, t, ǫ) can be asymptotically described using Whitham equations [37] and elliptic θ-functions.…”

“…The caustics separating the two regions consist of the leading edge, the trailing edge, and the point of gradient catastrophe itself, see Figure 1. Although we do not use this fact, we note that those borders are the curves on which the so-called Lax-Levermore maximizer [32,36] is singular and the associated Riemann surface changes from genus 0 to genus 1. The transitional asymptotic description of the KdV solution u(x, t, ǫ) for (x, t) approaching the caustics, has escaped investigation for a long time.…”

The thermodynamic limit of the Whitham equations

“…The wave dynamics problems leading to the modulation equations are typically connected with the study of the long-time evolution of large-scale initial data [3], [4] and involve the semiclassical asymptotics of the inverse scattering transform. In this asymptotics, the typical scale of modulations 1/ǫ is necessarily proportional to a global number of degrees of freedom in the problem g ≫ 1 (we assume no small parameters in the original equation, instead, we consider large-scale ∼ 1/ǫ initial data that can be approximated by gsoliton or g-gap potential).…”

The KdV Hierarchy: Universality and a Painlevé Transcendent

“…For t 1 < t c 1 , the KdV solution is approximated by the Hopf solution for small ǫ > 0, and for t 1 > t c 1 , an interval of rapid oscillations is formed where the KdV solution can be modeled using Jacobi elliptic θ-functions [26,31,39,11,12,24].…”