The KdV hierarchy: universality and a Painleve transcendent

2011/01/13 by Claeys, T., Grava, T.
#33E17 #35Q15 #35Q53 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1101.2602

Abstract

We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where \e→ 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 \e=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.

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