Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 045, 20 pages      arXiv:1412.2867      https://doi.org/10.3842/SIGMA.2015.045
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

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

Ognyan Christov a and 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

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

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.

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

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