Isomonodromy and Painlev\'e Type Equations, Case Studies

TL;DR

This paper explores isomonodromic families of linear ODEs with specific singularities, leading to classifications and new hierarchies of Painlevé-type equations, including rediscoveries and novel findings.

Contribution

It classifies Painlevé-type equations induced by isomonodromic families with specific singularities, revealing new hierarchies and rediscovering known results.

Findings

Identification of a small list of hierarchies of Painlevé-type equations.

Discovery of a likely new hierarchy related to classical P3(D8).

Presentation of a 'companion' to P1.

Abstract

There is an abundance of equations of Painlev\'e type besides the classical Painlev\'e equations. Classifications have been computed by the Japanese school. Here we consider Painlev\'e type equations induced by isomonodromic families of linear ODE's having at most ${z=0}$ and $z=\infty$ as singularities. Requiring that the formal data at the singularities produce isomonodromic families parametrized by a single variable $t$ leads to a small list of hierarchies of cases. The study of these cases involves Stokes matricesand moduli for linear ODE's on the projective line. Case studies reveal interesting families of linear ODE's and Painlev\'e type equations. However, rather often the complexity (especially of the Lax pair) is too high for either the computations or for the output. Apart from classical Painlev\'e equations one rediscovers work of Harnad, Noumi and Yamada. A hierarchy, probably new, related to the classical $P_3(D_8)$, is discovered. Finally, an amusing ''companion'' of $P_1$ is presented.