On the Linearization of the First and Second Painleve' Equations

TL;DR

This paper constructs new linear Fuchs--Garnier pairs for the first and second Painleve' equations, enabling transformations between different known matrix pairs and advancing the understanding of their linear representations.

Contribution

It introduces novel 3x3 matrix Fuchs--Garnier pairs for Painleve' equations and develops an invertible integral transform linking different 2x2 matrix pairs.

Findings

New 3x3 linear Fuchs--Garnier pairs for Painleve' equations

An invertible integral transform relating 2x2 matrix pairs

Unified framework connecting known Painleve' matrix pairs

Abstract

We found Fuchs--Garnier pairs in 3X3 matrices for the first and second Painleve' equations which are linear in the spectral parameter. As an application of our pairs for the second Painleve' equation we use the generalized Laplace transform to derive an invertible integral transformation relating two its Fuchs--Garnier pairs in 2X2 matrices with different singularity structures, namely, the pair due to Jimbo and Miwa and the one found by Harnad, Tracy, and Widom. Together with the certain other transformations it allows us to relate all known 2X2 matrix Fuchs--Garnier pairs for the second Painleve' equation with the original Garnier pair.