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Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

Title:

Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation

Bertola, Marco and Cafasso, M. (2012) Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation. Communications in Mathematical Physics, 309 (3). pp. 793-833. ISSN 0010-3616

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Official URL: http://dx.doi.org/10.1007/s00220-011-1383-x

Abstract

We extend the formalism of integrable operators à la Its-Izergin-Korepin-Slavnov to matrix-valued convolution operators on a semi–infinite interval and to matrix integral operators with a kernel of the form E T 1 (λ)E 2 (μ) λ+μ , thus proving that their resolvent operators can be expressed in terms of solutions of some specific Riemann-Hilbert problems. We also describe some applications, mainly to a noncommutative version of Painlevé II (recently introduced by Retakh and Rubtsov) and a related noncommutative equation of Painlevé type. We construct a particular family of solutions of the noncommutative Painlevé II that are pole-free (for real values of the variables) and hence analogous to the Hastings-McLeod solution of (commutative) Painlevé II. Such a solution plays the same role as its commutative counterpart relative to the Tracy–Widom theorem, but for the computation of the Fredholm determinant of a matrix version of the Airy kernel.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Article
Refereed:Yes
Authors:Bertola, Marco and Cafasso, M.
Journal or Publication:Communications in Mathematical Physics
Date:2012
Digital Object Identifier (DOI):10.1007/s00220-011-1383-x
ID Code:976935
Deposited By: DANIELLE DENNIE
Deposited On:05 Mar 2013 16:01
Last Modified:18 Jan 2018 17:43

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