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

Abstract

In this paper we study the equation $$w^{(4)}=5w"(w^2-w^{\prime })+5w(w^{\prime }{)}^2-w^5+(\lambda z+\alpha )w+\gamma ,$$ 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 $\gamma /\lambda =3k$, $\gamma /\lambda =3k-1$, $k\in \mathbb{Z}$, 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.

References in corpus (2)

• The Hamiltonian Structure of the Second Painleve Hierarchy
• Bäcklund Transformations for First and Second Painlevé Hierarchies