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

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 Schrödinger 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.

11 pages

References in corpus (2)

• Shock problem for MKdV equation: Long time Dynamics of the Step-like initial data
• Inverse scattering theory for Schrödinger operators with steplike potentials

Cited by in corpus (5)

• Rigorous asymptotics of a KdV soliton gas
• On the long time asymptotic behavior of the modified Korteweg de Vries equation with step-like initial data
• Soliton-mean field interaction in Korteweg-de Vries dispersive hydrodynamics
• Whitham equations and phase shifts for the Korteweg-deVries equation
• KdV equation beyond standard assumptions on initial data