The Third, Fifth and Sixth Painlev\'e Equations on Weighted Projective Spaces

TL;DR

This paper investigates the third, fifth, and sixth Painlevé equations using weighted projective spaces, providing a unified geometric framework that includes singular normal forms, initial condition spaces, Riccati solutions, and Boutroux's coordinates.

Contribution

It introduces a systematic approach to study Painlevé equations via weighted projective spaces, leveraging orbifold structures and dynamical systems theory for a unified analysis.

Findings

Unified geometric framework for Painlevé equations

Explicit descriptions of initial condition spaces

Analysis of Riccati solutions and Boutroux's coordinates

Abstract

The third, fifth and sixth Painlev\'e equations are studied by means of the weighted projective spaces ${\mathbb C}P^3(p,q,r,s)$ with suitable weights $(p,q,r,s)$ determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux's coordinates are systematically studied in a unified way with the aid of the orbifold structure of ${\mathbb C}P^3(p,q,r,s)$ and dynamical systems theory.