Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

Abstract

We consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$, holomorphically depending on parameter $t$ within a polydisk in ${\mathbb{C}}^n$ centered at $t=0$, such that the eigenvalues of the leading matrix at $z=\infty$ coalesce along a locus $\Delta$ contained in the polydisk, passing through $t=0$. Namely, $z=\infty$ is a resonant irregular singularity for $t\in \Delta$. We analyze the case when the leading matrix remains diagonalizable at $\Delta$. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as $t$ varies in the polydisk, and their limits for $t$ tending to points of $\Delta$. 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 polydisk (including $\Delta$) in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at the fixed coalescence point $t=0$. Conversely, when the system is isomonodromic in a small domain not intersecting $\Delta$ inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that $\Delta$ is not a branching locus for the $t$-continuation of fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius manifolds is explained. An application to Painlevé equations is discussed.