The KdV Hierarchy: Universality and a Painlevé Transcendent

We study the Cauchy problem for the Korteweg–de Vries (KdV) hierarchy in the small dispersion limit where ϵ→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 that corresponds to ϵ=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.