Metrisability of Painlevé equations

Abstract

We solve the metrisability problem for the six Painlevé equations, and more generally for all 2nd order ODEs with Painlevé property, and determine for which of these equations their integral curves are geodesics of a (pseudo) Riemannian metric on a surface.A problem of characterising metrisable ODEs by differential invariants was posed by Roger Liouville [21], who has reduced it to an overdetermined system of linear PDEs (see Theorem 2.1 in the next section). The complete solution was provided relatively recently [1], where it was shown that an ODE is metrisable if and only if three point invariants of differential orders five and six vanish, and certain genericity assumptions hold.A different approach was developed by Painlevé, Kowalevskaya and Gambier who studied 2nd order ODEs in the complex domain [24,16].Definition 1.2. The ODE y ′′ = R(x, y, y ′ ), where R is a rational function of y and y ′ has the Painlevé property (PP) if its movable singularities (i.e. singularities whose locations depend on the initial conditions) are poles.The solutions of equations with Painlevé property are single-valued thus giving rise to proper functions on C. There exists fifty canonical types of second order ODEs with PP

Lie point symmetries and ODEs passing the Painlevé test

“…In this Appendix we present the interesting relation between the results, contained in [14], from now on indicated as their results and our results.…”

“…After completing this manuscript (and publishing its preprint as arXiv: 1712.09811v1) we became aware of a preprint by Contatto and Dunajski [14]. They address and solve a different problem, namely which of the second order ODEs having the Painlevé property are metrisable (i.e.…”

“…This isü = F (s,u) for dimL = 1,ü = 1 s F (u) for dimL = 2, where for (14) F is some specific function. Alternatively, once the values of α, β, γ, δ have been specified, one can simply ask Maple to integrate (14). Indeed the Lie group procedure has been built into the routines for analytically solving ODEs.…”

“…The linearizing transformation is known [14,27,40,44,45] and involves elliptic integrals and solutions of the Heun equation.…”

“…DL thanks the CRM for the support during his stay in Montréal where this research has been carried out. We thank M. Dunajski for calling the preprint [14] to our attention and for very helpful (electronic) discussions. We thank E. Cheb-Terrab for helping us in running the Rif program.…”