Philip Boalch
The Klein solution to Painlev\'e's sixth equation
Preprint series:
ESI preprints
- MSC:
- 34A20 Differential equations in the complex domain, See Also
Abstract: We will describe a method for constructing explicit algebraic solutions to the
sixth Painlev\'e equation.
There are basically two steps:
First we explain how to construct finite braid group orbits of triples
of elements of $\SL_2(\IC)$ 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 Painlev\'e 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 $\SL_2(\IC)$.
Keywords: Painleve equations, isomonodromy, Klein simple group