A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be used to perform symmetry reduction. The method generalizes one presented in a recent publication for the case of ordinary difference equations. In turn, it ...

A new singular perturbation method based on the Lie symmetry group is presented to a system of difference equations. This method yields consistent derivation of a renormalization group equation which gives an asymptotic solution of the difference equation. The renormalization group equation is a Lie differential equation of a Lie group which leaves the system approximately invariant. For a 2-D ...

We introduce an approach based on moving frames for polygon recognition and symmetry detection. We present detailed algorithms for recognition of polygons modulo the special Euclidean, Euclidean, equi-affine, skewedaffine and similarity Lie groups, and explain the procedure for a generic Lie group. The time complexity of our algorithms is linear in the number of vertices and they are noise resi...

The applications of symmetry groups to problems arising in the calculus of variations have their origins in the late papers of Lie, e.g., [34], which introduced the subject of “integral invariants”. Lie showed how the symmetry group of a variational problem can be readily computed based on an adaptation of the infinitesimal method used to compute symmetry groups of differential equations. Moreo...

where ξ is a real-valued function and η is a 2 × 2 matrix complex-valued function, a Lie symmetry of system (1) if commutation relation [L,X] = R(x)L, (4) holds with some 2× 2 matrix function R(x) (for details, see, e.g., Ref. [3]). A simple computation shows that if X is a Lie symmetry of system (1), then an operator X + r(x)L with a smooth function r(x) is its Lie symmetry as well. Hence we c...

Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined ‘defining system’ of differential equations. The usual computer classification...

arising in several application [4]. Lie symmetry of BEq was found in [5], while the Q-conditional symmetry (i.e., non-classical symmetry [6]) was described in [7] and [8]. In the general case a wide list of Lie symmetries for DC equations of the form (1) is presented in [9]. A complete description of Lie symmetries, i.e., group classification of (1) has been done in [10]. The Q-conditional symm...