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

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 {\frac{E_1^T(\lambda) E_2(\mu)}{\lambda+\mu}} , 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.

Type de document :
Article dans une revue
Communications in Mathematical Physics, Springer Verlag, 2012, 309 (3), pp.793 - 833. 〈10.1007/s00220-011-1383-x〉
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https://hal.archives-ouvertes.fr/hal-01390757
Contributeur : Okina Université d'Angers <>
Soumis le : mercredi 2 novembre 2016 - 14:02:19
Dernière modification le : lundi 5 février 2018 - 15:00:03

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Marco Bertola, Mattia Cafasso. Fredholm Determinants and Pole-free Solutions to the Noncommutative Painlevé II Equation. Communications in Mathematical Physics, Springer Verlag, 2012, 309 (3), pp.793 - 833. 〈10.1007/s00220-011-1383-x〉. 〈hal-01390757〉

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