Isomonodromic differential equations and differential categories

TL;DR

This paper explores the concept of isomonodromicity in parameterized linear differential equations using differential categories, providing criteria that simplify testing for isomonodromicity without solving nonlinear equations.

Contribution

It establishes that isomonodromicity can be checked parameter-wise under certain conditions and links it to conjugacy of differential Galois groups, extending previous results.

Findings

Isomonodromicity is equivalent to parameter-wise isomonodromicity under filtered-linearly closed fields.

Testing isomonodromicity does not require solving nonlinear differential equations.

Isomonodromicity corresponds to conjugacy to constants of the differential Galois group.

Abstract

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters as we show by giving a counterexample. Also, we show that isomonodromicity is equivalent to conjugacy to constants of the associated parameterized differential Galois group, extending a result of P. Cassidy and M. Singer, which we also prove categorically. We illustrate our main results by a series of examples, using, in particular, a relation between Gauss-Manin connection and parameterized differential Galois groups.