Variations for Some Painlevé Equations

Abstract

This paper rst discusses irreducibility of a Painlev e equation P. We explain how the Painlev e 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 Painlev e 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 di erential Galois group for the rst two variational equations. As expected there are no cases where this group is commutative.