On a reduction of the generalized Darboux-Halphen system
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.
Cite
@article{arxiv.1710.00158,
title = {On a reduction of the generalized Darboux-Halphen system},
author = {Sumanto Chanda and Sarbarish Chakravarty and Partha Guha},
journal= {arXiv preprint arXiv:1710.00158},
year = {2018}
}
Comments
Some content of this article overlaps with content of another article written previously by two of the authors: arXiv:1606.02910. We have also removed the error, expanding and writing it into a completely new form, several new results having been found. 14 pages