Abstract
In this paper we are concerned with the integrability of the fifth Painleve equation ( ) from the point of view of the Ham iltonian dynamics. We prove that the Painleve equation (2) with parameters for arbitrary complex V P V 0 =0 , = (and more generally with parameters related by Baclund 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-Simo method yield to the non-integrable results.
Highlights
IntroductionThe six Painlevé equations ( PI PVI ) were introduced and first studied by Paul Painlevé [1] and his student B
The six Painlevé equations ( PI PVI ) were introduced and first studied by Paul Painlevé [1] and his student B.Gambier [2] who classified all the rational differential equations of the second order d2 y dt 2 =R t, y, dy dt free of movable critical points
We prove that the Painlevé V equation (2) with parameters = 0, 0 = for arbitrary complex is non integrable by means of meromorphic first integrals
Summary
The six Painlevé equations ( PI PVI ) were introduced and first studied by Paul Painlevé [1] and his student B. In [7] Morales-Ruiz raises the question about the integrability of the Painlevé transcendents as Hamiltonian systems. Later Morales-Ruiz in [8], and Stoyanova and Christov in [9] obtain a non-integrable result for Painlevé II family. 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. Studying the differential Galois group of the first and second variational equations along a particular rational non-equilibrium solution we can find nonintegrable results. It appears that the corresponding variational equations have an irregular singularity and new difficulty have to be overcome.
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