Universality of the break-up profile for the KdV equation in the small dispersion limit using the Riemann-Hilbert approach

TL;DR

This paper rigorously derives the universal asymptotic behavior of the KdV equation near the gradient catastrophe point in the small dispersion limit, revealing a connection to a higher order Painleve-type equation.

Contribution

It provides a rigorous proof of the universal asymptotic expansion near the critical point using Riemann-Hilbert techniques, confirming a conjecture by Dubrovin.

Findings

Asymptotic expansion near gradient catastrophe point derived

Sub-leading term described by a higher order Painleve-type equation

Universal behavior confirmed for Hamiltonian perturbations of hyperbolic equations

Abstract

We obtain an asymptotic expansion for the solution of the Cauchy problem for the Korteweg-de Vries (KdV) equation in the small dispersion limit near the point of gradient catastrophe (x_c,t_c) for the solution of the dispersionless equation. The sub-leading term in this expansion is described by the smooth solution of a fourth order ODE, which is a higher order analogue to the Painleve I equation. This is in accordance with a conjecture of Dubrovin, suggesting that this is a universal phenomenon for any Hamiltonian perturbation of a hyperbolic equation. Using the Deift/Zhou steepest descent method applied on the Riemann-Hilbert problem for the KdV equation, we are able to prove the asymptotic expansion rigorously in a double scaling limit.