2006.320: Picard and Chazy solutions to the Painlevé VI equation

2006.320: Marta Mazzocco (2001) Picard and Chazy solutions to the Painlevé VI equation. Mathematische Annalen, 321 (1). pp. 131-169. ISSN 1432-1807

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DOI: 10.1007/PL00004500

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.

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