An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Value distribution of solutions of Painlevé differential equations

19th September 2006

Author: Laine, Ilpo (University of Joensuu)

Abstract

We present a survey of the value distribution on non-rational solutions of Painlevé differential equations in terms of the Nevanlinna theory. We consider their growth and, in particular, their deficiencies and ramification indices with respect both to constant and to small moving targets. An important role here is played by the fact that the second main theorem of the Nevanlinna theory for Painlevé solutions reduces to an asymptotic equality. In addition, we make some remarks concerning the value distribution of solutions of higher order Painlevé equations. List of open problems will be presented as well.