Picard and Chazy solutions to the Painlevé VI equation

Mazzocco, Marta (2001) Picard and Chazy solutions to the Painlevé VI equation. Mathematische Annalen, 321 (1). pp. 131-169. ISSN 1432-1807

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Abstract

We study the solutions of a particular family of Painlevé VI equations with parameters b = g = 0, d = \frac12 and 2a = (2m-1)2, for 2m Î \mathbb Z. We show that in the case of half-integer m, all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0,1,¥ and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVIm equation for any m such that 2m Î \mathbb Z. As an application, we classify all the algebraic solutions. For m half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For m integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions.

Item Type: Article
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 32 Several complex variables and analytic spaces
MSC 2010, the AMS's Mathematics Subject Classification > 34 Ordinary differential equations
MSC 2010, the AMS's Mathematics Subject Classification > 35 Partial differential equations
MSC 2010, the AMS's Mathematics Subject Classification > 53 Differential geometry
Depositing User: Ms Lucy van Russelt
Date Deposited: 16 Aug 2006
Last Modified: 20 Oct 2017 14:12
URI: http://eprints.maths.manchester.ac.uk/id/eprint/529

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