Solving PVI by Isomonodromy Deformations

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withdrawn by the author himself, because some improvements are necessary.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