Non-commutative Painleve' equations and Hermite-type matrix orthogonal polynomials

Abstract

We study double integral representations of Christoffel-Darboux kernels associated with two examples of Hermite-type matrix orthogonal polynomials. We show that the Fredholm determinants connected with these kernels are related through the Its-Izergin-Korepin-Slavnov (IIKS) theory with a certain Riemann-Hilbert problem. Using this Riemann-Hilbert problem we obtain a Lax pair whose compatibility conditions lead to a non-commutative version of the Painleve' IV differential equation for each family.

Final version, accepted for publication on CMP: Communications in Mathematical Physics. 24 pages, 1 figure

Cited by in corpus (8)

• Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painleve VI equation
• On matrix Painlevé-4 equations. Part 1: Painlevé--Kovalevskaya test
• On matrix Painlevé-4 equations. Part 2: Isomonodromic Lax pairs
• Reducibility of Matrix Weights
• The Algebra of Differential Operators for a Gegenbauer Weight Matrix
• A fully noncommutative analog of the Painlevé IV equation and a structure of its solutions
• Ladder relations for a class of matrix valued orthogonal polynomials
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type