Regular flat structure and generalized Okubo system
arXiv:1702.03074
Abstract
We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlevé equations (except for PI). In our framework, the Painlevé equations PVI-PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlevé equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.
33 pages. arXiv admin note: text overlap with arXiv:1511.01608