# Isomonodromy Deformations at an Irregular Singularity with Coalescing Eigenvalues

## TL;DR

This paper studies how solutions to certain linear differential equations behave when eigenvalues coalesce at an irregular singularity, extending isomonodromic deformations across coalescence points and analyzing their implications for Frobenius manifolds and Painlevé equations.

## Contribution

It demonstrates conditions under which isomonodromic deformations extend through coalescing eigenvalues, ensuring well-defined monodromy data and solutions across singularities.

## Key findings

- Isomonodromic deformations can be extended to coalescence points under minimal conditions.
- Fundamental solutions and monodromy data remain well-defined across coalescence loci.
- Vanishing Stokes matrix entries prevent branching of solutions at coalescence points.

## Abstract

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisc in $\mathbb{C}^n$ centred at $t=0$. The eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisc, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyse the case when the leading matrix remains diagonalisable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as $t$ varies in the polydisc, and their limits for $t$ tending to points of $\Delta$. When the deformation is isomonodromic away from $\Delta$, it is well known that a fundamental matrix solution has singularities at $\Delta$. When the system also has a Fuchsian singularity at $z=0$, we show under minimal vanishing conditions on the residue matrix at $z=0$ that isomonodromic deformations can be extended to the whole polydisc, including $\Delta$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed $t=0$. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting $\Delta$, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that $\Delta$ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.