Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
Vernadsky National Library of Ukraine
Переглянути архів Інформація| Поле | Співвідношення | |
| Title |
Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
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| Creator |
Christov, O.
Georgiev, G. |
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| 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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| Date |
2019-02-13T16:22:44Z
2019-02-13T16:22:44Z 2015 |
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| Type |
Article
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| Identifier |
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 http://dspace.nbuv.gov.ua/handle/123456789/147104 |
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| Language |
en
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| Relation |
Symmetry, Integrability and Geometry: Methods and Applications
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| Publisher |
Інститут математики НАН України
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