TY - JOUR
T1 - The KdV hierarchy: universality and a Painleve transcendent
JF - International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099
Y1 - 2012
A1 - Tom Claeys
A1 - Tamara Grava
KW - Small-Dispersion limit
AB - 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.
PB - Oxford University Press
UR - http://hdl.handle.net/1963/6921
N1 - This article was published in "International Mathematics Research Notices, vol. 22 (2012) , page 5063-5099
U1 - 6902
U2 - Mathematics
U4 - 1
ER -