Symmetry groups of differential–difference equations and their compatibility

It is shown that the intrinsic determining equations of a given differential–difference equation (DDE) can be derived by the compatibility between the original equation and the intrinsic invariant surface condition. The ( 2 + 1 ) -dimensional Toda lattice, the special Toda lattice and the DD-KP equation serving as examples are used to illustrate this approach. Then, Bäcklund transformations of the ( 2 + 1 ) -dimensional DDEs including the special Toda lattice, the modified Toda lattice and the DD-KZ equation are presented by using the non-intrinsic direct method. In addition, the Clarkson–Kruskal direct method is developed to find similarity reductions of the DDEs.