Discrete derivatives and symmetries of difference equations

  title={Discrete derivatives and symmetries of difference equations},
  author={Decio Levi and J. Negro and Mariano A. del Olmo},
  journal={Journal of Physics A},
We show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable for finding the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation. 
Lie point symmetries and commuting flows for equations on lattices
Different symmetry formalisms for difference equations on lattices are reviewed and applied to perform symmetry reduction for both linear and nonlinear partial difference equations. Both Lie pointExpand
Lie Symmetries for Lattice Equations
Lie symmetries has been introduced by Sophus Lie to study differential equations. It has been one of the most efficient way for obtaining exact analytic solution of differential equations. Here weExpand
Discretization of nonlinear evolution equations over associative function algebras
Abstract A general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces, defined over a regular lattice of points, in such a way thatExpand
Discrete q-derivatives and symmetries of q-difference equations
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 properties considered for shift invariantExpand
Continuous symmetries of difference equations
Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program:Expand
Umbral calculus, difference equations and the discrete Schrödinger equation
In this paper, we discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. TheExpand
From symmetries to number theory
It is shown that the finite-operator calculus provides a simple formalism useful for constructing symmetry-preserving discretizations of quantum-mechanical integrable models. A related algebraicExpand
S I ] 2 1 D ec 2 00 0 Lie symmetries of difference equations 1
The discrete heat equation is worked out in order to illustrate the search of symmetries of difference equations. It is paid an special attention to the Lie structure of these symmetries, as well asExpand
Lorentz and Galilei Invariance on Lattices
We show that the algebraic aspects of Lie symmetries and generalized symmetries in nonrelativistic and relativistic quantum mechanics can be preserved in linear lattice theories. The mathematicalExpand
Heat Polynomials, Umbral Correspondence and Burgers Equations
We show that the umbral correspondence between differential equations can be achieved by means of a suitable transformation preserving the algebraic structure of the problems. We present the generalExpand


Continuous symmetries of discrete equations
Abstract Lie group techniques for solving differential equations are extended to differential-difference equations. As an application, it is shown that the two-dimensional Toda lattice has anExpand
Symmetries of the wave equation in a uniform lattice
The symmetries of discrete versions of a class of equations (that includes Klein - Gordon and wave equations) on a two-dimensional grid are studied. They close the same Lie algebra as theExpand
Symmetries of the heat equation on the lattice
Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties areExpand
Symmetries and conditional symmetries of differential difference equations
Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined withExpand
Lie group formalism for difference equations
The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples areExpand
Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction
Abstract A method is given to derive the point symmetries of partial differential-difference equations. Applying the method to the Kacvan Moerbeke equation we find its symmetries form aExpand
Transformation groups in net spaces
We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformationsExpand
Representation of Lie groups and special functions
At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, theExpand
The Similarity Method for Difference Equations
On applique la methode de similitude aux equations aux differences. Quand une equation aux differences a une dimension admet une symetrie, l'equation devient lineaire et on peut obtenir uneExpand
Applications of lie groups to differential equations
1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves andExpand