Symmetry Groups Of Differential Equations Research Articles

PROLONGED ANALYTIC CONNECTED GROUP ACTIONS ARE GENERICALLY FREE

We prove that an effective, analytic action of a connected Lie group G on an analytic manifold M becomes free on a comeager subset of an open subset of M when prolonged to a frame bundle of sufficiently high order. We further prove that the action of G becomes free on a comeager subset of an open subset of a submanifold jet bundle over M of sufficiently high order, thereby establishing a general result that underlies Lie's theory of symmetry groups of differential equations and the equivariant method of moving frames.

Algorithms for Differential Invariants of Symmetry Groups of Differential Equations

We develop new computational algorithms, based on the method of equivariant moving frames, for classifying the differential invariants of Lie symmetry pseudo-groups of differential equations and analyzing the structure of the induced differential invariant algebra. The Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP) equations serve to illustrate examples. In particular, we deduce the first complete classification of the differential invariants and their syzygies of the KP symmetry pseudo-group.

Moving frames and singularities of prolonged group actions

The prolongation of a transformation group to jet bundles forms the geometric foundation underlying Lie's theory of symmetry groups of differential equations, the theory of differential invariants, and the Cartan theory of moving frames. Recent developments in the moving frame theory have necessitated a detailed understanding of the geometry of prolonged transformation groups. This paper begins with a basic review of moving frames, and then focuses on the study of both regular and singular prolonged group orbits. Highlights include a corrected version of the basic stabilization theorem, a discussion of singular points, and geometric and algebraic characterizations of totally singular submanifolds, which are those that admit no moving frame. In addition to applications to the method of moving frames, the paper includes a generalized Wronskian lemma for vector-valued functions, and methods for the solution to Lie determinant equations.

Scale invariance and the Bohr–Wilson–Sommerfeld (BWS) quantization for power law one-dimensional potential wells

The classical periods of motion τ(E) are computed for a particle under the influence of a potential well of the form U(x)=α‖x‖ν, with both ν and α positive real constants. Assuming the reflection convention at the origin, these results can be extended to the cases where both ν and α are negative real constants. Also, the scale invariance exhibited by these potentials is analyzed using dimensional arguments directly on the classical equations of motion as well as the more powerful Lie method, appropriate for studying one-parameter symmetry groups of differential equations. The action variables J(E) are obtained from τ(E) and the Bohr–Wilson–Sommerfeld (BWS) quantization rule for the energy spectrum of all the above potentials is reobtained. An interpretation of the results is given in the light of semiclassical arguments.