EQUIVARIANT MOVING FRAME METHOD AND THE LOCAL EQUIVALENCE OF u_(xx) = r(x, u, v, u_x, v_x) UNDER FIBER-PRESERVING TRANSFORMATIONS

摘要

We use the equivariant moving frame method to study the local equivalence problem of scalar control equations of the form u_(xx) = r(x, u, v, u_x, v_x) under the pseudo-group of fiber-preserving transformations X = φ(x), U = β(x, u), V = α(x, u, v). Three typical branches of the equivalence problem are considered: the degenerate case, which contains the control systems with largest fiber-preserving symmetry group, the branch containing the Hilbert-Cartan equation, and the generic case.