Lie symmetry reductions and exact solutions of some differential–difference equations

Abstract

In this paper, a class of differential–difference equations (DDEs) are considered for Lie group analysis. With the help of symbolic computation MATHEMATICA, the continuous Lie point symmetry technique is extended to obtain corresponding infinitesimals. Similarity reductions are derived by solving the characteristic equations. Then some exact solutions are presented by using inverse transformations. In addition, starting from concrete realization of the generalized Virasoro type symmetry algebra [σ(f1), σ(f2)] = σ(f′1f2 − f1f′2), many high-dimensional DDEs can be derived. For example, we give out the (2+1)-dimensional Toda lattice, modified Toda lattice and special Toda lattice in a uniform way.