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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