Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
In this paper we study the equation
 w⁽⁴⁾=5w′′(w²−w′)+5w(w′)²−⁵+(λz+α)w+γ,
 which is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulatio...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2015 |
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Інститут математики НАН України
2015
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/147104 |
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| Zitieren: | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville / O. Christov, G. Georgiev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862741100334153728 |
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| author | Christov, O. Georgiev, G. |
| author_facet | Christov, O. Georgiev, G. |
| citation_txt | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville / O. Christov, G. Georgiev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In this paper we study the equation
w⁽⁴⁾=5w′′(w²−w′)+5w(w′)²−⁵+(λz+α)w+γ,
which is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulation, Bäcklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters γ/λ=3k, γ/λ=3k−1, k∈Z, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the PII-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.
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| first_indexed | 2025-12-07T20:18:29Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147104 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:18:29Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
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| spelling | Christov, O. Georgiev, G. 2019-02-13T16:22:44Z 2019-02-13T16:22:44Z 2015 Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville / O. Christov, G. Georgiev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 32 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 70H05; 70H07; 34M55; 37J30 DOI:10.3842/SIGMA.2015.045 https://nasplib.isofts.kiev.ua/handle/123456789/147104 In this paper we study the equation
 w⁽⁴⁾=5w′′(w²−w′)+5w(w′)²−⁵+(λz+α)w+γ,
 which is one of the higher-order Painlevé equations (i.e., equations in the polynomial class having the Painlevé property). Like the classical Painlevé equations, this equation admits a Hamiltonian formulation, Bäcklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters γ/λ=3k, γ/λ=3k−1, k∈Z, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the PII-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability. This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full
 collection is available at http://www.emis.de/journals/SIGMA/AMDS2014.html. 
 The authors are grateful to the referees for their constructive criticism and suggestions. We
 also would like to thank Ivan Dimitrov for many useful discussions. O.C. acknowledges partial
 support by Grant 059/2014 with Sofia University. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville Article published earlier |
| spellingShingle | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville Christov, O. Georgiev, G. |
| title | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville |
| title_full | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville |
| title_fullStr | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville |
| title_full_unstemmed | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville |
| title_short | Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville |
| title_sort | non-integrability of some higher-order painlevé equations in the sense of liouville |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147104 |
| work_keys_str_mv | AT christovo nonintegrabilityofsomehigherorderpainleveequationsinthesenseofliouville AT georgievg nonintegrabilityofsomehigherorderpainleveequationsinthesenseofliouville |