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

Tsvetana Stoyanova

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DOI: 10.4236/apm.2011.14031                

Abstract

In this paper we are concerned with the integrability of the fifth Painlevé equation (P_V ) 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.

Keywords

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

Conflicts of Interest

The authors declare no conflicts of interest.

References

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