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Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
Інший ID:
2010 Mathematics Subject Classification: 70H05; 70H07; 34M55; 37J30
DOI:10.3842/SIGMA.2015.045
DOI:10.3842/SIGMA.2015.045
Посилання:
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.
Підтримка:
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.
Дата:
2015
Переглядів:
438
Завантажень:
227
Анотація:
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.
