Article

# Solving PVI by Isomonodromy Deformations

06/2011;

Source: arXiv

**ABSTRACT** The critical and asymptotic behaviors of solutions of the sixth Painlev\'e

equation, an their parametrization in terms of monodromy data, are

synthetically reviewed. The explicit formulas are given. This paper has been

withdrawn by the author himself, because some improvements are necessary.

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**ABSTRACT:**We represent and analyze the general solution of the sixth Painlevé transcendent $ \mathcal{P}_6 $ \mathcal{P}_6 in the Picard-Hitchin-Okamoto class in the Painlevé form as the logarithmic derivative of the ratio of τ-functions. We express these functions explicitly in terms of the elliptic Legendre integrals and Jacobi theta functions, for which we write the general differentiation rules. We also establish a relation between the $ \mathcal{P}_6 $ \mathcal{P}_6 equation and the uniformization of algebraic curves and present examples.Theoretical and Mathematical Physics 12/2010; 161(3):1616-1633. DOI:10.1007/s11232-009-0150-z · 0.70 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the analytic properties and the critical behavior of the elliptic representation of solutions of the Painlev\'e 6 equation. We solve the connection problem for elliptic representation in the generic case and in a non-generic case equivalent to WDVV equations of associativity.Communications on Pure and Applied Mathematics 10/2002; 55(10):1280 - 1363. DOI:10.1002/cpa.10045 · 3.08 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We will describe a method for constructing explicit algebraic solutions to the sixth Painleve equation, generalising that of Dubrovin-Mazzocco. There are basically two steps: First we explain how to construct finite braid group orbits of triples of elements of SL_2(C) out of triples of generators of three-dimensional complex reflection groups. (This involves the Fourier-Laplace transform for certain irregular connections.) Then we adapt a result of Jimbo to produce the Painleve VI solutions. (In particular this solves a Riemann-Hilbert problem explicitly.) Each step will be illustrated using the complex reflection group associated to Klein's simple group of order 168. This leads to a new algebraic solution with seven branches. We will also prove that, unlike the algebraic solutions of Dubrovin-Mazzocco and Hitchin, this solution is not equivalent to any solution coming from a finite subgroup of SL_2(C). The results of this paper also yield a simple proof of a recent theorem of Inaba-Iwasaki-Saito on the action of Okamoto's affine D4 symmetry group as well as the correct connection formulae for generic Painleve VI equations.Proceedings of the London Mathematical Society 09/2003; 90. DOI:10.1112/S0024611504015011 · 1.12 Impact Factor

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