Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 088, 10 pages      arXiv:1705.07625      https://doi.org/10.3842/SIGMA.2019.088
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

Variations for Some Painlevé Equations

Primitivo B. Acosta-Humánez ab, Marius van der Put c and Jaap Top c
a) School of Basic and Biomedical Sciences, Universidad Simón Bolívar, Barranquilla, Colombia
b) Instituto Superior de Formación Docente Salomé Ureña - ISFODOSU, Santiago de los Caballeros, Dominican Republic
c) Bernoulli Institute, University of Groningen, Groningen, The Netherlands

Received November 01, 2018, in final form November 05, 2019; Published online November 09, 2019

Abstract
This paper first discusses irreducibility of a Painlevé equation $P$. We explain how the Painlevé property is helpful for the computation of special classical and algebraic solutions. As in a paper of Morales-Ruiz we associate an autonomous Hamiltonian $\mathbb{H}$ to a Painlevé equation $P$. Complete integrability of $\mathbb{H}$ is shown to imply that all solutions to $P$ are classical (which includes algebraic), so in particular $P$ is solvable by ''quadratures''. Next, we show that the variational equation of $P$ at a given algebraic solution coincides with the normal variational equation of $\mathbb{H}$ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases $P_{2}$ to $P_{5}$ where algebraic solutions are present, by showing how our results lead to a quick computation of the component of the identity of the differential Galois group for the first two variational equations. As expected there are no cases where this group is commutative.

Key words: Hamiltonian systems; variational equations; Painlevé equations; differential Galois groups.

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