Applied Mathematics Seminar

The famous Painlevé equations belong to the class y'' + a_1 {y'}^3 + 3 a_2 {y'}^2 + 3 a_3 y' + a_4 = 0, where a_i=a_i(x,y). All integrable equations in this class (with a_i rational in y) were classified by Painlevé/Gambier about 100 years ago. This class of equations is invariant under the general point transformation x=Phi(X,Y), y=Psi(X,Y) and it is therefore very difficult to find out whether two equations in this class are related. We describe R. Liouville's theory of invariants that can be used to construct invariant characteristic expressions (syzygies), and in particular present such characterizations for Painlevé equations I-IV. Using them one can quickly determine whether a given equation has any chance of being trasformed into a Painlevé equation. The forms of the syzygies could also be used for reorganizing the Gambier/Ince list in a more systematic and invariant way.

This work was done in collaboration with V. Dryuma.