Preliminary group classification of the nonlinear differential–difference equations

Preliminary group classification of the nonlinear differential–difference equations of the form u ̈ n = F n ( t , u n − 1 , u n , u n + 1 ) is considered by using the classical infinitesimal Lie method and the theory of classification of abstract low-dimensional Lie algebras. Restrict to Lie point symmetries, and assume that the Lie algebra of the symmetry group is realized by a particular vector field. It is proved that there is only one equation admitting three-dimensional simple Lie algebras. What is more, all the inequivalent equations admitting semi-simple Lie algebras are nothing but the one. Also there exist inequivalent equations admitting low-dimensional solvable Lie algebras and semi-direct sums of semi-simple and solvable Lie algebras.