Special Solutions of the Sixth Painleve Equation with Solvable Monodromy

TL;DR

This paper investigates special, non-classical solutions of the sixth Painleve equation that are invariant under symmetry transformations, providing exact monodromy calculations and geometric characterizations on Fricke's cubic surface.

Contribution

It introduces a new class of special solutions invariant under Backlund symmetries, extending understanding beyond classical solutions with explicit monodromy analysis.

Findings

Exact linear monodromy of these solutions is computed.

Solutions are characterized on Fricke's cubic surface.

Most fixed points of symmetries are classical solutions, but these are not.

Abstract

We will study two types of special solutions of the sixth Painleve equation, which are invariant under the symmetries obtained from the Backlund transformations. In most cases, the fixed points of the Backlund transformations are classical solution, but our solutions are not classical for generic parameters. We will calculate the linear monodromy of these solutions exactly, and we will characterize them on Fricke's cubic surface of monodromy.