2004
DOI: 10.1112/s0024611504015011
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From Klein to Painlevé via Fourier, Laplace and Jimbo

Abstract: Abstract. We will describe a method for constructing explicit algebraic solutions to the sixth Painlevé 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 Painlevé VI solutions. (In par… Show more

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Cited by 105 publications
(235 citation statements)
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“…The logarithm of this rate will give the entropy of the Poincaré return map γ * . Recently several authors [4,5,9,13,14,25,26] have been interested in finding algebraic solutions, which must have only finitely many branches under the analytic continuations along all loops in Z. On the other hand, in this article we will be concerned with those solutions which are finitely many-valued along a fixed single loop.…”
Section: Introductionmentioning
confidence: 99%
“…The logarithm of this rate will give the entropy of the Poincaré return map γ * . Recently several authors [4,5,9,13,14,25,26] have been interested in finding algebraic solutions, which must have only finitely many branches under the analytic continuations along all loops in Z. On the other hand, in this article we will be concerned with those solutions which are finitely many-valued along a fixed single loop.…”
Section: Introductionmentioning
confidence: 99%
“…We note that the parametrization that was adopted in [29] to write system (3.9) explicitly in terms of y where y(t) is a solution of P 6 is not unique. Alternate parameterizations have been identified by Boalch [3]- [4] in his studies of P 6 . Since this system can be mapped to the irregular 3 × 3 Lax pair of [18] and [29] via the generalized Laplace transform, see equation (3.12) above, it follows that these parameterizations are equivalent to system (3.9) up to a gauge transformation.…”
Section: Similarity Solution Of 3wri System In Terms Of the Sixth Paimentioning
confidence: 99%
“…The Fuchs-Garnier pairs associated with each Painlevé equation play a very important role in the theory and applications of the Painlevé equations. Nowadays, as a result of the intensive studies of the Painlevé equations, many different Fuchs-Garnier pairs have been derived [3,4,12,18,20,22,24,26,27,30,32,33,41]. The methods and ideas used in these derivations vary widely.…”
Section: Introductionmentioning
confidence: 99%
“…Here we list the relevant results in the asymptotics of Painlevé VI as studied by [28] and [46] and listed in [52]. The problem was also considered in [53].…”
Section: A Schlesinger System Asymptotics and Painlevé VImentioning
confidence: 99%
“…The dependence on φ has been considered in a number of papers [28,46], and is explicit at the Painlevé singular points t = 0, 1, ∞. Let us take t i → 1 as the asymptotic point for definiteness.…”
Section: Jhep07(2014)132mentioning
confidence: 99%