The KdV Hierarchy: Universality and a Painlevé Transcendent

Abstract

We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where ǫ → 0. For negative analytic initial data with a single negative hump, we prove that for small times, the solution is approximated by the solution to the hyperbolic transport equation which corresponds to ǫ = 0. Near the time of gradient catastrophe for the transport equation, we show that the solution to the KdV hierarchy is approximated by a particular Painlevé transcendent. This supports Dubrovins universality conjecture concerning the critical behavior of Hamiltonian perturbations of hyperbolic equations. We use the Riemann-Hilbert approach to prove our results.

On critical behaviour in generalized Kadomtsev–Petviashvili equations

“…It has been conjectured in [13] and proved in [6] (and for the KdV hierarchy in [7]) that near the point (x c , t c ) the solution of KdV in the limit → 0 is approximated by…”