# Lie Symmetries for Lattice Equations

@inproceedings{Levi2004LieSF, title={Lie Symmetries for Lattice Equations}, author={Decio Levi}, year={2004} }

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 we show how one can extend this technique to the case of differential difference and difference equations.

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#### References

SHOWING 1-10 OF 68 REFERENCES

Continuous symmetries of discrete equations

- Physics
- 1991

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 an… Expand

Lie point symmetries of difference equations and lattices

- Mathematics, Physics
- 2000

A method is presented for finding the Lie point symmetry transformations acting simultaneously on difference equations and lattices, while leaving the solution set of the corresponding difference… Expand

Lie group formalism for difference equations

- Mathematics
- 1997

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 are… Expand

Lie symmetries and the integration of difference equations

- Physics
- 1993

Abstract The Lie method is generalised to ordinary difference equations. We prove that the order of ordinary difference equations can be reduced by one, provided the equation under consideration… Expand

Dilation symmetries and equations on the lattice

- Mathematics
- 1999

We discuss the role of dilation symmetries for differential difference equations depending on nearest-neighbour interactions. In particular, we show that for a simple class of differential difference… Expand

Symmetries and conditional symmetries of differential difference equations

- Mathematics
- 1993

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 with… Expand

Discrete derivatives and symmetries of difference equations

- Mathematics, Physics
- 2001

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… Expand

Lie symmetries of finite‐difference equations

- Physics
- 1995

Discretizations of the Helmholtz, heat, and wave equations on uniform lattices are considered in various space–time dimensions. The symmetry properties of these finite‐difference equations are… Expand

Symmetries of discrete dynamical systems involving two species

- Mathematics, Physics
- 1999

The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in… Expand

Non-point integrable symmetries for equations on the lattice

- Mathematics
- 2000

We present a new class of non-point groups of transformations for scalar evolution chain equations. Then we construct the class of differential equations on the lattice which admits such group… Expand