Differential Invariants of Second Order ODEs, I

Abstract

This paper is devoted to differential invariants of equations y = a 3 (x, y)y 3 + a 2 (x, y)y 2 + a 1 (x, y)y + a 0 (x, y).w.r.t. point transformations. The natural bundle of these equations and its bundles of kjets of sections, k = 0, 1, 2, . . . , are considered. The action of the pseudogroup of all point transformations on these bundles is investigated. Tensor differential invariants distinguishing orbits of this action on jet bundles of second and third orders are constructed. A complete collection of generators and their differential syzygies is obtained for the algebra of all scalar differential invariants of a wide class of considered equations containing generic equations.

Open Problems for Painlevé Equations

“…where F is rational in w and dw/dz and locally analytic in z and solved the equations in terms of the first, second and fourth Painlevé transcendents, elliptic functions, or quadratures. For various results on classifying classes of second-order ordinary differential equations, including Painlevé equations, see Babich and Bordag [12], Bagderina [13,15,16,17,18], Bagderina and Tarkhanov [19], Berth and Czichowski [22], Hietarinta and Dryuma [78], Kamran, Lamb and Shadwick [88], Kartak [90,91,92,93], Kossovskiy and Zaitsev [97], Milson and Valiquette [106], Valiquette [139] and Yumaguzhin [145]. Most of these studies are concerned with the invariance of second-order ordinary differential equations of the form…”

Poincaré Function for Moduli of Differential-Geometric Structures

“…This action is transitive in 2-jets, and transitive outside the stratum F 3 = 0 in 3-jets, where F 3 is the Liouville relative invariant [29], see also [16]. Differential invariants of this action were counted in [39,44]: h k = 0 for k < 4, h k = 2(k − 1) for k ≥ 4. Therefore we obtain…”

Differential invariants and operators of invariant differentiation of the projectable action of Lie groups

“…Many particular results were then obtained for concrete groups and systems of equations [5]- [9]. Significant progress in the theory of invariants was recently achieved by the efforts of Olver [10], [11].…”