Duke Mathematical Journal

Isomonodromy deformations at an irregular singularity with coalescing eigenvalues

Giordano Cotti, Boris Dubrovin, and Davide Guzzetti

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Abstract

We consider an n×n linear system of ODEs with an irregular singularity of Poincaré rank 1 at z=, holomorphically depending on parameter t within a polydisk in Cn centered at t=0, such that the eigenvalues of the leading matrix at z= coalesce along a locus Δ contained in the polydisk, passing through t=0. Namely, z= is a resonant irregular singularity for tΔ. We analyze the case when the leading matrix remains diagonalizable at Δ. 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 Δ. 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 Δ) 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 Δ inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that Δ 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.

Article information

Source
Duke Math. J., Volume 168, Number 6 (2019), 967-1108.

Dates
Received: 28 September 2017
Revised: 28 September 2018
First available in Project Euclid: 13 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1552442775

Digital Object Identifier
doi:10.1215/00127094-2018-0059

Mathematical Reviews number (MathSciNet)
MR3934594

Zentralblatt MATH identifier
07055221

Subjects
Primary: 34M56: Isomonodromic deformations
Secondary: 34M35: Singularities, monodromy, local behavior of solutions, normal forms 34M40: Stokes phenomena and connection problems (linear and nonlinear)

Keywords
isomonodromic deformations singularities monodromy local behavior of solutions normal forms Stokes’ matrices nonadmissible deformations coalescence of eigenvalues Frobenius manifolds Painleve’ equations

Citation

Cotti, Giordano; Dubrovin, Boris; Guzzetti, Davide. Isomonodromy deformations at an irregular singularity with coalescing eigenvalues. Duke Math. J. 168 (2019), no. 6, 967--1108. doi:10.1215/00127094-2018-0059. https://projecteuclid.org/euclid.dmj/1552442775


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