Point equivalence of second-order ODEs: Maximal invariant classification order

Abstract

We show that the local equivalence problem of second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also demonstrate that, modulo Cartan duality and point transformations, the Painlevé-I equation can be characterized as the simplest second-order ordinary differential equation belonging to the class of equations requiring 10th order jets for their classification. Formalization of the problem and resultsFollowing standard practices, [12,25], we let p = u x , q = u xx denote the first and second order derivatives of a single variable function u = u(x). Then, two second-order ordinary differential equations q = f (x, u, p) and Q = F (X, U, P ), p = u x , q = u xx , P = U X , Q = U XX ,

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…”

Notes on Lie symmetry group methods for differential equations

“…This theorem states that for second-order ordinary differential equations Eq. y 2 = f (x, y, p) with p = y 1 is point equivalent to the trivial free-particle equation y 2 = 0 if and only if the following fourth-order Tresse (absolute) invariants [25] I 1 = f pppp = 0, I 2 = D 2 x f pp − 4 D x f yp − f p D x f pp + 6f yy − 3f y f pp + 4f p f yp = 0, (121) are identically zero. Here D x = ∂ x + p∂ y + f ∂ p is the truncation of the usual total derivative operator D x .…”

Solving Local Equivalence Problems with the Equivariant Moving Frame Method

“…In Case II), it can be shown, see [24] for more details, that when Q P 2 X 2 ≡ 0 the (partially normalized) lifted invariants Q P 3 X 3 and Q P 2 X 4 cannot be simultaneously equal to zero. There are then 2 subbundles of regular submanifold jets…”