From Klein to Painlevé Via Fourier, Laplace and Jimbo

By P. Boalch. 2003

149 citations recorded by the source corpus.

We describe a method for constructing explicit algebraic solutions to the sixth Painlevé equation, generalising that of Dubrovin and Mazzocco. There are basically two steps. First we explain how to construct finite braid group orbits of triples of elements of SL2(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 particular, this solves a Riemann–Hilbert problem explicitly.)

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/045c48eb06b6bc4fe205a56fe52cdd473615752c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. O. Lisovyy
2. Y. Tykhyy
3. K. Corlette
4. Carlos Simpson

In Mathematical Physics

1. Carlos Simpson
2. O. Lisovyy
3. Y. Tykhyy
4. K. Corlette
5. Alexander V. Kitaev

In Statistical and Nonlinear Physics

1. O. Lisovyy
2. Y. Tykhyy
3. K. Corlette
4. Carlos Simpson
5. Alexander V. Kitaev

Citations in the corpus

• Katz's middle convolution algorithm (2006, unknown): “A G ] 2 5 O ct”
• Algebraic solutions of the sixth Painleve equation (2008, background): “Boalch [2] refers to an equation equivalent to (10) as ‘Fricke relation’.”
• On the classification of rank-two representations of quasiprojective fundamental groups (2007, background): “Such solutions have been constructed by Hitchin [56], Dubrovin [43], Boalch [ 15 ], Ben Hamed-Gavrilov [12].”