Dynamics of the Sixth Painlevé Equation

Abstract

The sixth Painlevé equation is hiding extremely rich geometric structures behind its outward appearance. This article tries to give as a total picture as possible of its dynamical natures, based on the Riemann-Hilbert approach recently developed by the authors, using various techniques from algebraic geometry. A good part of the contents is extended to Garnier systems, while this article is restricted to the original sixth Painlevé equation.

56 pages, 18 figures, reference updated

References in corpus (3)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
• Nodal curves and Riccati solutions of Painlevé equations

Cited by in corpus (7)

• Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I
• Surface Operators and Knot Homologies
• Holomorphic dynamics, Painlevé VI equation and Character Varieties
• Classification of algebraic solutions of irregular Garnier systems
• Normal forms for rank two linear irregular differential equations and moduli spaces
• Periodic Solutions to Painlevé VI and Dynamical System on Cubic Surface
• Finite branch solutions to Painleve VI around a fixed singular point