Variations for Some Painlevé Equations

Abstract

This paper first discusses irreducibility of a Painleve equation P . We show that reducibility is equivalent to the existence of special classical 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 special classical or algebraic, so in particular P is reducible.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 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.

Nonintegrability of the Painlevé IV equation in the Liouville–Arnold sense and Stokes phenomena

“…Their approach is based on Liouville's theory of elementary functions and some properties of elliptic integrals. Acosta-Humánez et al 1 claim that when 𝑎 = 0, 𝑏 = − 2 9…”

“…𝐻 (𝑥), … the derivatives of 𝑋 𝐻 (𝑥) with respect to 𝑥. By successive derivation of (6) with respect to 𝑧 and evaluation at 𝑧 0 , we obtain the so-called 𝑘 variational equations of order 𝑘 (VE) 𝑘 for the function 𝑥(𝑡) ẋ(𝑘) (𝑡) = 𝑋 (1) 𝐻 (𝑥(𝑡)) 𝑥 (𝑘) (𝑡) + 𝑃 ( 𝑥 (1) (𝑡), 𝑥 (2) (𝑡), … , 𝑥 (𝑘−1) (𝑡) ) .…”

“…Their approach is based on Liouville's theory of elementary functions and some properties of elliptic integrals. Acosta‐Humánez et al 1 . claim that when $$a=0, b=-\frac{2}{9}$$, and $$a=0, b=-2$$ the Painlevé IV equation is not integrable as a Hamiltonian system.…”

Classical solutions of the degenerate fifth Painlevé equation

“…These solutions of the Painlevé equations are often called 'classical solutions', see [45,46]. It is well known that solutions of deg-P V (1) are related to solutions of the third Painlevé equation…”

“…In section 2, the relationship between deg-P V (1) and the third Painlevé equation ( 4) is discussed using the associated Hamiltonian. In section 3, classical solutions of the third Painlevé equation (4) are reviewed, the rational solutions in section 3.1 and the Bessel function solutions in section 3.2.…”