A real pole-free solution to a higher order Painlevé I equation
by
Tom Claeys
K.U.Leuven
Coauthors: Maarten Vanlessen

We consider a fourth order ODE, which is an analogue to the Painlevé I equation. We prove the existence of a real pole-free solution to this equation, which was conjectured by Dubrovin, using Riemann-Hilbert methods. This existence follows from the solvability of a certain Riemann-Hilbert problem, which can be proven using a so-called vanishing lemma. The pole-free solution plays a role in critical random matrix ensembles, and describes asymptotics in the small dispersion limit of the KdV equation near the point of gradient catastrophe.

Date received: June 19, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavg-14.