The fifty-two icosahedral solutions to Painlevé VI
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Philip Boalch
Abstract
The solutions of the (nonlinear) Painlevé VI differential equation having icosahedral linear monodromy group will be classified up to equivalence under Okamoto's affine F4 Weyl group action and many properties of the solutions will be given.
There are 52 classes, the first ten of which correspond directly to the ten icosahedral entries on Schwarz's list of algebraic solutions of the hypergeometric equation. The next nine solutions are simple deformations of known PVI solutions (and have less than five branches) and five of the larger solutions are already known, due to work of Dubrovin and Mazzocco and Kitaev.
Of the remaining 28 solutions we will find 20 explicitly using the method of [P. P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. 90 (2005), no. 3, 167–208.] (via Jimbo's asymptotic formula). Amongst those constructed there is one solution that is “generic” in that its parameters lie on none of the affine F4 hyperplanes, one that is equivalent to the Dubrovin-Mazzocco elliptic solution and three elliptic solutions that are related to the Valentiner three-dimensional complex reflection group, the largest having 24 branches.
© Walter de Gruyter
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- Mild pro-p-groups and Galois groups of p-extensions of ℚ
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Articles in the same Issue
- Free boundary regularity below the continuous threshold: 2-phase problems
- Lp-theory of the Navier-Stokes flow in the exterior of a moving or rotating obstacle
- The Kirillov-Reshetikhin conjecture and solutions of T-systems
- Hyperbolicity of nodal hypersurfaces
- Unipotent orbits and local L-functions
- Circular sets of prime numbers and p-extensions of the rationals
- The μ-invariant of anticyclotomic L-functions of imaginary quadratic fields
- Zum Hauptsatz von C. Jordan über ganzzahlige Darstellungen endlicher Gruppen
- Mild pro-p-groups and Galois groups of p-extensions of ℚ
- The fifty-two icosahedral solutions to Painlevé VI
- Residue currents of holomorphic morphisms
