Ognyan Christov, Georgi Georgiev, “Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville”, SIGMA, 2015, Volume 11,045, 20 pp.

This article is cited in 5 papers

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

Ognyan Christov^a, Georgi Georgiev^b

^a Faculty of Mathematics and Informatics, Sofia University, 1164 Sofia, Bulgaria
^b Department of Mathematics and Informatics, University of Transport, 1574, Sofia, Bulgaria

Abstract: 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.

Keywords: Painlevé type equations; Hamiltonian systems; differential Galois groups; generalized confluent hypergeometric equations.

MSC: 70H05; 70H07; 34M55; 37J30

Received: December 10, 2014; in final form June 10, 2015; Published online June 17, 2015

Language: English

DOI: 10.3842/SIGMA.2015.045