Quadratic Differential Systems and Chazy Equations, I

Details Quadratic Differential Systems and Chazy Equations, I Quadratic Differential Systems and Chazy Equations, I

2012-03-01

arxiv.org/abs/1203.0283v2

Mathematics

Classical Analysis and ODEs

68 pages, 1 figure; slightly expanded

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Quadratic Differential Systems and Chazy Equations, I does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Quadratic Differential Systems and Chazy Equations, I, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Quadratic Differential Systems and Chazy Equations, I will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-165086