Non-integrability of Painlevé VI equations in the Liouville sense

Abstract

In this paper we consider the sixth Painlevé equation as a Hamiltonian system depending on parameters. We prove the non-integrability of two families of the Painlevé VI equations, i.e. non-existence of meromorphic first integrals. For this we utilize the differential Galois theory and the Ziglin-Ramis-Morales-Ruiz method.

Non-Integrability of Painlevé V Equations in the Liouville Sense and Stokes Phenomenon

“… VI equation for some particular values of the parameters is proved by Horozov and Stoyanova in [10] and by Stoyanova in [11]. In the present note we continue the study of Painlevé transcendents with the fifth Painlevé equation and obtain an analogous result for one family of the parameters.…”

On linear systems and τ functions associated with Lamé's equation and Painlevé's equation VI

“…First we introduce a linear system for the differential equation (3.4); later we introduce a linear system that realises the kernel most naturally associated with P V I . For notational simplicity, we often suppress the dependence of operators upon t. The following result is a consequence of results of Turrittin [31,27], who clarified certain facts about the Birkhoff canonical form for matrices.…”

The holonomy group at infinity of the Painlevé VI equation

“…This theory asserts that integrability in a Liouville sense along a particular solution Γ c implies that the variational equation E 1 , as well all higher order variational equations E k along this solution, have virtually commutative differential Galois groups. Indeed, in such a way the "semi-local" non-integrability in a neighborhood of some particular solutions and parameter values of the PVI system has been recently proved by Horozov and Stoyanova [7,16], see also Morales-Ruiz [12]. To prove the non-integrability for all parameters we need, however, an explicitly known particular solution which exists for all parameter values.…”

“…is a double ramified covering over λ = 0, 1, c, ∞. Denote the monodromy matrix of the second variational equation(16) along α by T α . The entries of the monodromy matrix T α along a closed loop α on the elliptic surface S c are quadratic polynomials in the complete elliptic integrals of first and second kind along α.…”