Variations for Some Painlevé Equations
This paper first discusses the irreducibility of a Painlevé equation . We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz, we associate an autonomous Hamiltonian ℍ to a Painlevé equation . Complete integrabi...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2019 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2019
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/210300 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Variations for Some Painlevé Equations / P.B. Acosta-Humánez, M. van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862621342904352768 |
|---|---|
| author | Acosta-Humánez, P.B. van der Put, M. Top, J. |
| author_facet | Acosta-Humánez, P.B. van der Put, M. Top, J. |
| citation_txt | Variations for Some Painlevé Equations / P.B. Acosta-Humánez, M. van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | This paper first discusses the irreducibility of a Painlevé equation . We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz, we associate an autonomous Hamiltonian ℍ to a Painlevé equation . Complete integrability of ℍ is shown to imply that all solutions to are classical (which includes algebraic), so in particular is solvable by "quadratures". Next, we show that the variational equation of at a given algebraic solution coincides with the normal variational equation of ℍ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases ₂ to ₅ where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected, there are no cases where this group is commutative.
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| first_indexed | 2025-12-07T21:25:04Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-210300 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T21:25:04Z |
| publishDate | 2019 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Acosta-Humánez, P.B. van der Put, M. Top, J. 2025-12-05T09:26:36Z 2019 Variations for Some Painlevé Equations / P.B. Acosta-Humánez, M. van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E17; 34M55 arXiv: 1705.07625 https://nasplib.isofts.kiev.ua/handle/123456789/210300 https://doi.org/10.3842/SIGMA.2019.088 This paper first discusses the irreducibility of a Painlevé equation . We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz, we associate an autonomous Hamiltonian ℍ to a Painlevé equation . Complete integrability of ℍ is shown to imply that all solutions to are classical (which includes algebraic), so in particular is solvable by "quadratures". Next, we show that the variational equation of at a given algebraic solution coincides with the normal variational equation of ℍ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases ₂ to ₅ where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected, there are no cases where this group is commutative. We thank the referees of an earlier version of this paper for their useful suggestions. The first-named author thanks the Universidad Simon Bolivar and the Bernoulli Institute of Groningen University for the financial support of his research visit during which the initial version of this paper was written. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Variations for Some Painlevé Equations Article published earlier |
| spellingShingle | Variations for Some Painlevé Equations Acosta-Humánez, P.B. van der Put, M. Top, J. |
| title | Variations for Some Painlevé Equations |
| title_full | Variations for Some Painlevé Equations |
| title_fullStr | Variations for Some Painlevé Equations |
| title_full_unstemmed | Variations for Some Painlevé Equations |
| title_short | Variations for Some Painlevé Equations |
| title_sort | variations for some painlevé equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210300 |
| work_keys_str_mv | AT acostahumanezpb variationsforsomepainleveequations AT vanderputm variationsforsomepainleveequations AT topj variationsforsomepainleveequations |