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

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 $γ / λ = 3 k$ , $\gamma/\lambda = 3 k - 1 $,$ k \in $Z$ $, 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 $P_{I I}$ -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.

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 $γ / λ = 3 k$ , $\gamma/\lambda = 3 k - 1 $,$ k \in $Z$ $, 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 $P_{I I}$ -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.

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