On the form of dispersive shock waves of the Korteweg-de Vries equation

TL;DR

This paper analyzes the long-time behavior of solutions to the Korteweg-de Vries shock problem, showing it can be described as a modulated elliptic solution with explicit spectral data depending on initial conditions and scattering data.

Contribution

It provides a detailed description of the dispersive shock wave structure for the KdV equation, explicitly relating the spectral parameters to initial data and scattering information.

Findings

Long-time solutions are described as modulated one-gap elliptic solutions.

The elliptic modulus depends only on initial step size and propagation direction.

The phase shift is explicitly computed via the Jacobi inversion problem.

Abstract

We show that the long-time behavior of solutions to the Korteweg-de Vries shock problem can be described as a slowly modulated one-gap solution in the dispersive shock region. The modulus of the elliptic function (i.e., the spectrum of the underlying Schr\"odinger operator) depends only on the size of the step of the initial data and on the direction, $\frac{x}{t}=const.$, along which we determine the asymptotic behavior of the solution. In turn, the phase shift (i.e., the Dirichlet spectrum) in this elliptic function depends also on the scattering data, and is computed explicitly via the Jacobi inversion problem.