Variations for Some Painlevé Equations

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.

References in corpus (5)

• Order one equations with the Painlevé property
• Galoisian Methods for Testing Irreducibility of Order Two Nonlinear Differential Equations
• Moduli spaces for linear differential equations and the Painlevé equations
• Isomonodromy for the Degenerate Fifth Painlevé Equation
• A note on the R. Fuchs's problem for the Painlevé equations

Cited by in corpus (1)

• Classical Solutions of the Degenerate Fifth Painlevé Equation