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: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Integrability of Difference Equations Through Algebraic Entropy and Generalized Symmetries

Given an equation arising from some application or theoretical consideration one of the first questions one might ask is: What is its behavior? It is integrable? In these lectures we will introduce two different ways for establishing (and in some sense also defining) integrability for difference equations: Algebraic Entropy and Generalized Symmetries. Algebraic Entropy deals with the degrees of growth of the solution of any kind of discrete equation (ordinary, partial or even differential-difference) and usually provides a quick test to establish if an equation is or not integrable. The approach based on Generalized Symmetries also provides tools for investigating integrable equations and to find particular solutions by symmetry reductions. The main focus of the lectures will be on the computational tools that allow us to calculate Generalized Symmetries and extract the value of the Algebraic Entropy from a finite number of iterations of the map