Continuous symmetries of difference equations.

Abstract

We consider the study of symmetry analysis of difference equations. The original work done by

Lie about a century ago is known to be one of the best methods of solving differential equations.

Lie's theory of difference equations on the contrary, was only first explored about twenty years

ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some

work has been done in the field of symmetries of difference equations for the past years. Given

an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to

analyze the problem. It is commonly known that the number of independent variables can be

reduced after the symmetries of the equation are obtained. One can determine the optimal

system of the equation in order to get a reduction of the independent variables. In addition,

using the method, one can obtain new solutions from known ones. This feature is interesting

because some differential equations have apparently useless trivial solutions, but applying Lie

symmetries to them, more interesting solutions are obtained.

The question arises when it happens that our equation contains a discrete quantity. In other

words, we aim at investigating steps to be performed when we have a difference equation. Doing

so, we find symmetries of difference equations and use them to linearize and reduce the order

of difference equations. In this work, we analyze the work done by some researchers in the field

and apply their results to some examples.

This work will focus on the topical review of symmetries of difference equations and going

through that will enable us to make some contribution to the field in the near future.