A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants
We consider Fredholm determinants of matrix Hankel operators associated with matrix versions of the -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 partic...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211186 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants. Sofia Tarricone. SIGMA 17 (2021), 002, 25 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862718010660225024 |
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| author | Tarricone, Sofia |
| author_facet | Tarricone, Sofia |
| citation_txt | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants. Sofia Tarricone. SIGMA 17 (2021), 002, 25 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider Fredholm determinants of matrix Hankel operators associated with matrix versions of the -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.
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| first_indexed | 2026-03-20T18:10:42Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211186 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-20T18:10:42Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Tarricone, Sofia 2025-12-25T13:25:37Z 2021 A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants. Sofia Tarricone. SIGMA 17 (2021), 002, 25 pages 1815-0659 2020 Mathematics Subject Classification: 34M56; 35Q15; 47B35; 33C10 arXiv:2007.05707 https://nasplib.isofts.kiev.ua/handle/123456789/211186 https://doi.org/10.3842/SIGMA.2021.002 We consider Fredholm determinants of matrix Hankel operators associated with matrix versions of the -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. This project was supported by the European Union Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE 2017 grant agreement no. 778010 IPaDEGAN. I am also grateful to my supervisors, Mattia Cafasso and Marco Bertola, for their useful advice and guidance. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants Article published earlier |
| spellingShingle | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants Tarricone, Sofia |
| title | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants |
| title_full | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants |
| title_fullStr | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants |
| title_full_unstemmed | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants |
| title_short | A Fully Noncommutative Painlevé II Hierarchy: Lax Pair and Solutions Related to Fredholm Determinants |
| title_sort | fully noncommutative painlevé ii hierarchy: lax pair and solutions related to fredholm determinants |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211186 |
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