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6 - Generalized Lie Symmetries for Difference Equations

Published online by Cambridge University Press:  05 July 2011

By
Decio Levi  and
Ravil I. Yamilov
Edited by
Decio Levi ,
Peter Olver ,
Zora Thomova  and
Pavel Winternitz
Decio Levi
Affiliation:
Università degli Studi Roma Tre and Sezione INFN Roma Tre
Ravil I. Yamilov
Affiliation:
Ufa Institute of Mathematics
Decio Levi
Affiliation:
Università degli Studi Roma Tre
Peter Olver
Affiliation:
University of Minnesota
Zora Thomova
Affiliation:
SUNY Institute of Technology
Pavel Winternitz
Affiliation:
Université de Montréal
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Summary

Introduction

In this chapter we discuss the application of generalized symmetries to the investigation of difference and differential-difference equations. This is a sequel to the presentation of P. Winternitz where Lie point symmetries for difference equations have been introduced and studied in detail. In particular it has been shown there that for a given discrete equation, unless we allow for variable lattices, i.e., we consider a difference scheme, very few symmetries are present. So, if we want to get symmetries for difference equations, either we consider the point symmetries of a difference scheme or we extend the class of symmetries to the case of the generalized symmetries. In the following we will proceed in this second direction and analyze the structure of the generalized symmetries for a difference equation. We will limit ourselves to consider just partial difference equations (with two independent variables) where the lattice is fixed and non-transformable and either all independent variables are discrete (n,m) or one is discrete n and one is continuous t. We will limit our discussion to the case of scalar equations of a low order, i.e., when the dependent variable is a scalar and the differential difference equations involve at most derivatives of the second order of the fields and nearest neighboring interactions.

Type
Chapter
Information
Symmetries and Integrability of Difference Equations , pp. 160 - 190
Publisher: Cambridge University Press
Print publication year: 2011

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References

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