Variations for Some Painlevé Equations

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Authors: Acosta-Humánez, P.B., van der Put, M., Top, J.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210300
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Variations for Some Painlevé Equations / P.B. Acosta-Humánez, M. van der Put, J. Top // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 32 назв. — англ.

This paper first discusses the 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 ℍ to a Painlevé equation 𝘗. Complete integrability of 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 ℍ at the corresponding solution. Finally, we test the Morales-Ramis theorem in all cases 𝘗₂ to 𝘗₅ 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.