On a reduction of the generalized Darboux-Halphen system

TL;DR

This paper studies a generalized Darboux-Halphen system derived from self-dual Yang-Mills equations, transforming it into a constrained dynamical system, and provides solutions using hypergeometric functions.

Contribution

It introduces a new reduction of the generalized Darboux-Halphen system to a constrained dynamical system with explicit solutions.

Findings

The system can be transformed into a non-autonomous dynamical system.

The reduced system admits a Lax pair and Hamiltonian formulation.

Solutions are expressed in terms of hypergeometric functions.

Abstract

The equations for the general Darboux-Halphen system obtained as a reduction of the self-dual Yang-Mills can be transformed to a third-order system which resembles the classical Darboux-Halphen system with a common additive terms. It is shown that the transformed system can be further reduced to a constrained non-autonomous, non-homogeneous dynamical system. This dynamical system becomes homogeneous for the classical Darboux-Halphen case, and was studied in the context of self-dual Einstein's equations for Bianchi IX metrics. A Lax pair and Hamiltonian for this reduced system is derived and the solutions for the system are prescribed in terms of hypergeometric functions.