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P. Painlevé, “Sur les équations Différentielles du Second Ordre et d’Ordre Supérieur dont l’Intégrale Générale est Uniforme,” Acta Mathematica, Vol. 25, No. 1, December 1902, pp. 1-86.

has been cited by the following article:

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

  AUTHORS: Tsvetana Stoyanova

  KEYWORDS: Differential Galois theory, Painlevé V equation, Hamiltonian Systems, Stokes Phenomenon, Asymptotic Theory

  JOURNAL NAME: Advances in Pure Mathematics, Vol.1 No.4, July 27, 2011

  ABSTRACT: In this paper we are concerned with the integrability of the fifth Painlevé equation (PV ) from the point of view of the Hamiltonian dynamics. We prove that the PainlevéV equation (2) with parameters k∞=0,k0= –θ for arbitrary complex θ (and more generally with parameters related by Bäclund transformations) is non integrable by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and Ziglin-Ramis-Morales-Ruiz-Simó method yield to the non-integrable results.