Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations
In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/214185 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations. Malik Balogoun and Marco Bertola. SIGMA 21 (2025), 097, 50 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862687021446725632 |
|---|---|
| author | Balogoun, Malik Bertola, Marco |
| author_facet | Balogoun, Malik Bertola, Marco |
| citation_txt | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations. Malik Balogoun and Marco Bertola. SIGMA 21 (2025), 097, 50 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions.
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| first_indexed | 2026-03-17T13:21:48Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-214185 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T13:21:48Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Balogoun, Malik Bertola, Marco 2026-02-20T07:56:48Z 2025 Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations. Malik Balogoun and Marco Bertola. SIGMA 21 (2025), 097, 50 pages 1815-0659 2020 Mathematics Subject Classification: 33E17; 34M55; 33C47 arXiv:2411.08853 https://nasplib.isofts.kiev.ua/handle/123456789/214185 https://doi.org/10.3842/SIGMA.2025.097 In this paper, we analyze the asymptotic behaviour of the poles of certain rational solutions of the fifth Painlevé equation. These solutions are constructed by relating the corresponding tau function to a Hankel determinant of a certain sequence of moments. This approach was also used by one of the authors and collaborators in the study of the rational solutions of the second Painlevé equation. More specifically, we study the roots of the corresponding polynomial tau function, whose location corresponds to the poles of the associated rational solution. We show that, upon suitable rescaling, the roots asymptotically fill a region bounded by analytic arcs when the degree of the polynomial tau function tends to infinity and the other parameters are kept fixed. Moreover, we provide an approximate location of these roots within the region in terms of suitable quantization conditions. The second author completed the work during his tenure as Royal Society Wolfson Visiting Fellow (RSWVF/R2/242024) at the School of Mathematics in Bristol University. The work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC)grant RGPIN-2023-04747. Both authors thank the anonymous referees for the detailed reports and bibliographic improvements. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations Article published earlier |
| spellingShingle | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations Balogoun, Malik Bertola, Marco |
| title | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations |
| title_full | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations |
| title_fullStr | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations |
| title_full_unstemmed | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations |
| title_short | Rational Solutions of Painlevé V from Hankel Determinants and the Asymptotics of Their Pole Locations |
| title_sort | rational solutions of painlevé v from hankel determinants and the asymptotics of their pole locations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214185 |
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