Symbolic Computation and Lie Symmetry Groups for Two Nonlinear Differential-Difference Equations

Based on the symbolic computation system-Maple, the symmetry group direct method is extended to investigate Lie symmetry groups of two differential-difference equations. Through analysis and tedious calculation, the full symmetry groups of the well-known $\mathrm{D}\Delta$-KP equation and Toda lattice equation are obtained. From them, both the Lie point symmetry groups and a group of discrete transformations can be obtained. Furthermore, based on the full symmetry groups and some simple solutions of these two equations, some general solutions are constructed.