Variations for Some Painlevé Equations

Abstract

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.