Different linearizations of non-abelian second Painlev\'e systems and related monodromy surfaces

TL;DR

This paper explores different linearizations of non-abelian second Painlevé systems, establishing correspondences and deriving non-abelian monodromy surfaces with associated Poisson structures, advancing understanding of their integrable structures.

Contribution

It introduces a method to connect various linearizations of non-abelian Painlevé systems and derives their monodromy surfaces, including Poisson structures, which is a novel extension of classical theory.

Findings

Established HTW-JM correspondence method

Derived non-abelian monodromy surfaces

Discussed Poisson structures for these surfaces

Abstract

In this paper, we discuss a connection between different linearizations for non-abelian analogs of the second Painlev\'e equation. For each of the analogs, we listed the pairs of the Harnard-Tracy-Widom (HTW), Flaschka-Newell (FN), and Jimbo-Miwa (JM) types. A method for establishing the HTW-JM correspondence is suggested. For one of the non-abelian analogs, we derive the corresponding non-abelian generalizations of the monodromy surfaces related to the FN- and JM-type linearizations. A natural Poisson structure associated with these monodromy surfaces is also discussed.