%0 Journal Article %J International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099 %D 2012 %T The KdV hierarchy: universality and a Painleve transcendent %A Tom Claeys %A Tamara Grava %K Small-Dispersion limit %X We study the Cauchy problem for the Korteweg-de Vries (KdV) hierarchy in the small dispersion limit where $\e\to 0$. 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 Painlev\'e 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. %B International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099 %I Oxford University Press %G en %U http://hdl.handle.net/1963/6921 %1 6902 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2013-05-29T08:08:55Z No. of bitstreams: 1 1101.2602v1.pdf: 327994 bytes, checksum: 5b5bbc9f9b74ce97c2bf5ae794698819 (MD5)