Weights, Kovalevskaya exponents and the Painlev\'{e} property

TL;DR

This paper explores the relationship between weighted degrees, Kovalevskaya exponents, and the Painlevé property, classifying regular weights and linking them to Painlevé equations and their solutions using singularity and dynamical systems theory.

Contribution

It introduces a classification of regular weights related to Painlevé equations and establishes a correspondence between Laurent series solutions and stable manifolds in quasihomogeneous Hamiltonian systems.

Findings

Each polynomial Painlevé equation has a regular weight.

Existence of differential equations with Painlevé property for each regular weight in 2- and 4-dimensional cases.

Stable manifolds correspond to Laurent series solutions in quasihomogeneous Hamiltonian systems.

Abstract

Weighted degrees of quasihomogeneous Hamiltonian functions of the Painlev\'{e} equations are investigated. A tuple of positive integers, called a regular weight, satisfying certain conditions related to singularity theory is classified. Each polynomial Painlev\'{e} equation has a regular weight. Conversely, for $2$ and $4$-dim cases, it is shown that there exists a differential equation satisfying the Painlev\'{e} property associated with each regular weight. Kovalevskaya exponents of quasihomogeneous Hamiltonian systems are also investigated by means of regular weights, singularity theory and dynamical systems theory. It is shown that there is a one-to-one correspondence between Laurent series solutions and stable manifolds of the associated vector field obtained by the blow-up of the system. For $4$-dim autonomous Painlev\'{e} equations, the level surface of Hamiltonian functions can be decomposed into a disjoint union of stable manifolds.