Painlev\'e scheme

TL;DR

This paper reviews the Painlevé scheme of the sixth Painlevé equation, exploring accessible singular points and local indices in Hirzebruch surfaces, and extends the analysis to related Painlevé systems and higher-dimensional differential systems.

Contribution

It introduces a Painlevé scheme framework using the Painlevé α-method, enabling the recovery of Painlevé VI and other systems, and studies higher-dimensional systems with symmetric group symmetries.

Findings

Painlevé scheme characterized by accessible singular points.

Recovery of Painlevé VI system from the scheme.

Extension to higher-dimensional systems with symmetric group symmetry.

Abstract

In this note, we review the notion of Painlev\'e scheme of the sixth Painlev\'e equation from the viewpoint of accessible singular point and its local index in the Hirzebruch surface of degree two ${\Sigma_2}$. The key method is Painlev\'e $\alpha$-method for each accessible singular point. Giving a Painlev\'e scheme in the differential system satisfying certain conditions, we can recover the Painlev\'e VI system with the polynomial Hamiltonian. We also consider the case of the Painlev\'e V,IV and III systems, respectively. Finally, we study non-linear ordinary differential systems in dimension two with only simple accessible singular $(n+2)$-points in the Hirzebruch surface of degree $n$; ${\Sigma_n}$. This equation has symmetry of symmetric group of degree $n+2$.