Algebraic solutions of the sixth Painlevé equation

DOI10.1016/j.geomphys.2014.05.010zbMath1307.34135arXiv0809.4873OpenAlexW2964276714MaRDI QIDQ741115

Yuriy Tykhyy, Oleg Lisovyy

Publication date: 10 September 2014

Published in: Journal of Geometry and Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0809.4873

zbMATH Keywords

Painlevé equationscharacter varietiesisomonodromy deformations

Mathematics Subject Classification ID

Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies (34M55) Isomonodromic deformations for ordinary differential equations in the complex domain (34M56)

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Cites Work

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• Quadratic transformations for the sixth Painlevé equation
• Monodromy problem and the boundary condition for some Painlevé equations
• Dynamics on character varieties and Malgrange irreducibility of Painlevé VI equation
• Monodromy of certain Painlevé-VI transcendents and reflection groups
• Transformations \(RS_4^2(3)\) of the ranks \(\leq 4\) and algebraic solutions of the sixth Painlevé equation
• A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation.
• An ergodic study of Painlevé VI
• Finite branch solutions to Painlevé VI around a fixed singular point
• Rational solutions of the Painlevé VI equation
• Some explicit solutions to the Riemann–Hilbert problem
• Towards a nonlinear Schwarz's list
• The fifty-two icosahedral solutions to Painlevé VI
• Higher Genus Icosahedral Painleve Curves
• A LECTURE ON THE OCTAHEDRON
• From Klein to Painlevé via Fourier, Laplace and Jimbo
• Fredholm determinants, Jimbo‐Miwa‐Ueno τ‐functions, and representation theory
• On Algebraic Solutions to Painleve VI
• Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions
• Grothendieck’s dessins d’enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations
• On Painlevé VI transcendents related to the Dirac operator on the hyperbolic disk
• Picard and Chazy solutions to the Painlevé VI equation

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