A fully noncommutative Painlev\'e II hierarchy: Lax pair and solutions related to Fredholm determinants

TL;DR

This paper introduces a fully noncommutative Painlevé II hierarchy linked to Fredholm determinants of matrix convolution operators, providing explicit Lax pairs and solutions using integrable operator theory and Riemann-Hilbert methods.

Contribution

It develops a novel noncommutative Painlevé II hierarchy with explicit Lax pairs and solutions expressed via Fredholm determinants, extending integrable systems theory.

Findings

Established a connection between matrix Airy functions and a noncommutative Painlevé II hierarchy.

Derived explicit Lax pairs for hierarchy members using Riemann-Hilbert analysis.

Expressed solutions in terms of Fredholm determinants of matrix Airy convolution operators.

Abstract

We consider Fredholm determinants of matrix convolution operators associated to matrix versions of the $n - $th Airy functions. Using the theory of integrable operators, we relate them to a fully noncommutative Painlev\'e II hierarchy, defined through a matrix valued version of the Lenard operators. In particular, the Riemann-Hilbert technique 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 explicitely written in terms of these matrix valued Lenard operators and some solution of the hierarchy are written in terms of Fredholm determinants of the square of the matrix Airy convolution operators.