Duke Mathematical Journal
- Duke Math. J.
- Volume 168, Number 6 (2019), 967-1108.
Isomonodromy deformations at an irregular singularity with coalescing eigenvalues
We consider an linear system of ODEs with an irregular singularity of Poincaré rank 1 at , holomorphically depending on parameter within a polydisk in centered at , such that the eigenvalues of the leading matrix at coalesce along a locus contained in the polydisk, passing through . Namely, is a resonant irregular singularity for . We analyze the case when the leading matrix remains diagonalizable at . We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon, and monodromy data as varies in the polydisk, and their limits for tending to points of . When the system also has a Fuchsian singularity at , we show, under minimal vanishing conditions on the residue matrix at , that isomonodromic deformations can be extended to the whole polydisk (including ) 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 . Conversely, when the system is isomonodromic in a small domain not intersecting inside the polydisk, we give certain vanishing conditions on some entries of the Stokes matrices, ensuring that is not a branching locus for the -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.
Duke Math. J., Volume 168, Number 6 (2019), 967-1108.
Received: 28 September 2017
Revised: 28 September 2018
First available in Project Euclid: 13 March 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 34M56: Isomonodromic deformations
Secondary: 34M35: Singularities, monodromy, local behavior of solutions, normal forms 34M40: Stokes phenomena and connection problems (linear and nonlinear)
isomonodromic deformations singularities monodromy local behavior of solutions normal forms Stokes’ matrices nonadmissible deformations coalescence of eigenvalues Frobenius manifolds Painleve’ equations
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