A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants - Archive ouverte HAL Access content directly
Journal Articles Symmetry, Integrability and Geometry : Methods and Applications Year : 2021

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

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Sofia Tarricone
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#### 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.

#### Domains

Mathematics [math]

### Dates and versions

hal-03115034 , version 1 (19-01-2021)

### Identifiers

• HAL Id : hal-03115034 , version 1
• ARXIV :
• DOI :

### Cite

Sofia Tarricone. A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants. Symmetry, Integrability and Geometry : Methods and Applications, 2021, ⟨10.3842/SIGMA.2021.002⟩. ⟨hal-03115034⟩

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