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Эта публикация цитируется в 5 научных статьях (всего в 5 статьях)
Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville
Ognyan Christova, Georgi Georgievb a Faculty of Mathematics and Informatics, Sofia University, 1164 Sofia, Bulgaria
b Department of Mathematics and Informatics, University of Transport, 1574, Sofia, Bulgaria
Аннотация:
In this paper we study the equation
$$
w^{(4)} = 5 w'' (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma,
$$
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 $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, 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 $\mathrm{P}_{\mathrm{II}}$-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.
Ключевые слова:
Painlevé type equations; Hamiltonian systems; differential Galois groups; generalized confluent hypergeometric equations.
Поступила: 10 декабря 2014 г.; в окончательном варианте 10 июня 2015 г.; опубликована 17 июня 2015 г.
Образец цитирования:
Ognyan Christov, Georgi Georgiev, “Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville”, SIGMA, 11 (2015), 045, 20 pp.
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/sigma1026 https://www.mathnet.ru/rus/sigma/v11/p45
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