Non-Integrability of Some Higher-Order Painlev\'e Equations in the Sense of Liouville

TL;DR

This paper investigates the non-integrability of a specific higher-order Painlevé equation using differential Galois theory, demonstrating it lacks Liouville integrability and extending the analysis to related equations in the hierarchy.

Contribution

It proves the non-integrability of a particular higher-order Painlevé equation in the Liouville sense using the Ziglin-Morales-Ruiz-Ramis approach, and analyzes related hierarchy equations.

Findings

The equation is not integrable in Liouville sense for certain parameter values.

The second and third hierarchy members have non-commutative Galois groups, obstructing integrability.

The analysis extends to generalized confluent hypergeometric equations.

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 Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e property). Like the classical Painlev\'e equations, this equation admits a Hamiltonian formulation, B\"acklund 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.