Related papers: Discrete derivatives and symmetries of difference …
We illustrate some simple ideas that can be used for obtaining a classification of small-dimensional solvable Lie algebras.Using these we obtain the classification of 3 and 4 dimensional solvable Lie algebras (over fields of any…
Symmetry is a powerful tool for finding analytical solutions to differential equations, both partial and ordinary, via the similarity variables or via the invariance of the equation under group transformations. It is the largest group of…
A symmetric function of $N$ variables can be given in terms of symmetric polynomials of these variables. We determine those symmetric polynomials in which the dual differential operators take the neatest form when expressed in terms of our…
We apply Lie symmetry analysis of partial differential equations (PDEs) to the Euler-Lagrange equations of the two-Higgs-doublet model (2HDM), to determine its scalar Lie point symmetries. A Lie point symmetry is a structure-preserving…
The aim of this paper is to study symmetries of linearly singular differential equations, namely, equations that can not be written in normal form because the derivatives are multiplied by a singular linear operator. The concept of…
Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equation. We apply this principle by finding dilatations and…
We study the notion of the Lie-holomorph of a Leibniz algebra, recently introduced by N. P. Souris as a generalisation of the classical holomorph construction for Lie algebras. We establish a connection between the Lie-holomorph…
A Mathematica based program has been elaborated in order to determine the symmetry group of a finite difference equation, by means of its differential representation. The package provides functions which enable us to solve the determining…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum…
In this note, we derive a Leibniz rule for difference quotient.
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
We develope a difference calculus analogous to the differential geometry by translating the forms and exterior derivatives to similar expressions with difference operators, and apply the results to fields theory on the lattice [Ref. 1]. Our…
A discrete version of Lagrangian reduction is developed in the context of discrete time Lagrangian systems on $G\times G$, where $G$ is a Lie group. We consider the case when the Lagrange function is invariant with respect to the action of…
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of first degree for systems of discrete evolution equations and an answer to why there exist…
Based on an original classification of differential equations by types of regular Lie group actions, we offer a systematic procedure for describing partial differential equations with prescribed symmetry groups. Using a new powerful…
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class…
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the…