Quadratic Differential Systems and Chazy Equations, I

TL;DR

This paper introduces generalized Darboux-Halphen systems, explores their rational solution maps, connects solutions to generalized Schwarzian equations, and classifies those with the Painleve property, providing explicit integrations for some cases.

Contribution

It extends the class of quadratic differential systems, establishes their connection to Schwarzian equations and hypergeometric functions, and classifies systems with the Painleve property.

Findings

Classification of proper non-DH gDH systems with the Painleve property.

Rational morphisms between gDH systems derived from hypergeometric transformations.

Explicit solutions of some gDH systems in elementary and elliptic functions.

Abstract

Generalized Darboux-Halphen (gDH) systems, which form a versatile class of three-dimensional homogeneous quadratic differential systems (HQDS's), are introduced. They generalize the Darboux-Halphen (DH) systems considered by other authors, in that any non-DH gDH system is affinely but not projectively covariant. It is shown that the gDH class supports a rich collection of rational solution-preserving maps: morphisms that transform one gDH system to another. The proof relies on a bijection between (i) the solutions with noncoincident components of any `proper' gDH system, and (ii) the solutions of a generalized Schwarzian equation (gSE) associated to it, which generalizes the Schwarzian equation (SE) familiar from the conformal mapping of hyperbolic triangles. The gSE can be integrated parametrically in terms of the solutions of a Papperitz equation, which is a generalized Gauss hypergeometric equation. Ultimately, the rational gDH morphisms come from hypergeometric transformations. A complete classification of proper non-DH gDH systems with the Painleve property (PP) is also carried out, showing how some are related by rational morphisms. The classification follows from that of non-SE gSE's with the PP, due to Garnier and Carton-LeBrun. As examples, several non-DH gDH systems with the PP are integrated explicitly in terms of elementary and elliptic functions.