Keldysh Institute of Applied Mathematics,
4 Miusskaya Pl.,
125047 Moscow,
E-mail: dorod@spp.keldysh.ru

Difference equations: symmetries, exact solutions, conservation laws

A review of several applications of the Lie groups of point transformations to  difference  equations,  meshes  (lattices),  difference functionals is presented. Some  examples of difference models (i.e. difference equations and  appropriate  meshes) which admit the same symmetry group as there continuous limits. In particularly there were constructed the whole set of difference models for second order ODEs, which posses symmetry group. For intagrable cases the discrete representations of ODEs  ("an  exact finite-difference scheme") were developed. The invariant variational problems for difference equations are considered.  The Lagrangian formalism and the Noether's type constructions  for  difference  fuctionals, meshes and difference equations are illustrated by several examples.