Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 002, 25 pages      arXiv:2007.05707      https://doi.org/10.3842/SIGMA.2021.002

A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants

Sofia Tarricone ab
a) LAREMA, UMR 6093, UNIV Angers, CNRS, SFR Math-Stic, France
b) Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec, Canada, H3G 1M8

Received July 25, 2020, in final form December 31, 2020; Published online January 05, 2021

Abstract
We consider Fredholm determinants of matrix Hankel operators associated to matrix versions of the $n$-th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlevé II hierarchy, defined through a matrix-valued version of the Lenard operators. In particular, the Riemann-Hilbert techniques used to study these integrable operators allows to find a Lax pair for each member of the hierarchy. Finally, the coefficients of the Lax matrices are explicitly written in terms of the matrix-valued Lenard operators and some solutions of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy Hankel operators.

Key words: Painlevé II hierarchy; Airy Hankel operator; Riemann-Hilbert problem; Lax pairs.

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