Results 1 
7 of
7
Moving frames and singularities of prolonged group actions
 Selecta Math. (N.S
"... Abstract. 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 ..."
Abstract

Cited by 18 (12 self)
 Add to MetaCart
Abstract. 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 “totally 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 vectorvalued functions, and methods for the solution to Lie determinant equations.
Quasiexact solvability and the direct approach to invariant subspaces
 J. Phys. A: Math. Gen
"... We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasiexactly solvable quantum Hamiltonians on t ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
We propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of sl(2). This approach is used here to construct new exactly solvable and quasiexactly solvable quantum Hamiltonians on the line which are not Liealgebraic. It is also applied to generate potentials with multiple algebraic sectors. We discuss two illustrative examples of these two applications: an interesting generalization of the Lamé potential which posses four algebraic sectors, and a quasiexactly solvable deformation of the Morse potential which is not Liealgebraic. 1
Higher Conditional Symmetry and Reduction of Initial Value Problems
"... We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for thirdorder evolution equations that admit reduction to Cauchy problems for syste ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We give the exposition of a generalized symmetry approach to reduction of initial value problems for nonlinear evolution equations in one spatial variable. Using this approach we classify the initial value problems for thirdorder evolution equations that admit reduction to Cauchy problems for systems of two ordinary di#erential equations. These reductions are shown to correspond to higher conditional symmetries admitted by the corresponding nonlinear evolution equations.
Ordinary Differential Operators Possessing Invariant Subspaces of the Power Type
"... Ordinary dierential operators possessing invariant subspaces spanned by the functions x , i = 0; 1; : : : ; n 1, are considered. A complete description of operators of the submaximal order n 2 is obtained. The dimension C 2n 1 of the linear space of such operators is conjectured to be the u ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Ordinary dierential operators possessing invariant subspaces spanned by the functions x , i = 0; 1; : : : ; n 1, are considered. A complete description of operators of the submaximal order n 2 is obtained. The dimension C 2n 1 of the linear space of such operators is conjectured to be the upper bound for the operators possessing arbitrary ndimensional invariant subspace. A general representation for translationinvariant operators is found.
STRUCTURE THEOREMS FOR LINEAR AND NONLINEAR DIFFERENTIAL OPERATORS ADMITTING INVARIANT POLYNOMIAL SUBSPACES
, 2006
"... Abstract. In this paper we derive structure theorems that characterize the spaces of linear and nonlinear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. In this paper we derive structure theorems that characterize the spaces of linear and nonlinear differential operators that preserve finite dimensional subspaces generated by polynomials in one or several variables. By means of the useful concept of deficiency, we can write explicit basis for these spaces of differential operators. In the case of linear operators, these results apply to the theory of quasiexact solvability in quantum mechanics, specially in the multivariate case where the Lie algebraic approach is harder to apply. In the case of nonlinear operators, the structure theorems in this paper can be applied to the method of finding special solutions of nonlinear evolution equations by nonlinear separation of variables. AMS subject classification: 47F5, 35K55, 81R15 1.
On the classification of conditionally integrable evolution systems in (1 + 1) dimensions
, 2007
"... We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)dimensional evolution equations of order n that admit an Nshock type solution with N ≤ n + 1. To this end we develop a refinement of the technique from our earlier work [12] where we completely characterized all (1+1 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)dimensional evolution equations of order n that admit an Nshock type solution with N ≤ n + 1. To this end we develop a refinement of the technique from our earlier work [12] where we completely characterized all (1+1)dimensional evolution systems ut = F(x,t,u,∂u/∂x,..., ∂ n u/∂x n) that are conditionally invariant under a given generalized (Lie–Bäcklund) vector field Q(x,t,u,∂u/∂x,..., ∂ k u/∂x k)∂/∂u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasiexactly solvable models in quantum mechanics.
unknown title
, 2008
"... Geometric interplay between function subspaces and their rings of differential operators ..."
Abstract
 Add to MetaCart
Geometric interplay between function subspaces and their rings of differential operators