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

Abstract

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.