The Painlev\'{e} I hierarchy: Correspondence between the isomonodromic approach and the minimal models of the KP hierarchy

TL;DR

This paper establishes a detailed correspondence between the isomonodromic and minimal models approaches to the Painlevé I hierarchy, providing new explicit formulas for Lax matrices and Hamiltonians.

Contribution

It introduces an explicit link between two different formalisms for the Painlevé I hierarchy, enhancing understanding and computational tools.

Findings

Explicit correspondence between isomonodromic and minimal models

New expressions for Lax matrices

New formulas for Hamiltonians

Abstract

Two approaches to the Painlev\'{e} I hierarchy are discussed: the isomonodromic construction based on meromorphic connections, and the minimal models construction based on a reduction of the KP hierarchy. An explicit correspondence between both formalisms is built which gives the identification of these setups. In particular, this provides new expressions for the Lax matrices and Hamiltonians.