The Group Analysis Of Some Differential-Difference Equations

In this paper, some differential-difference equations are considered by using Lie group method. The symmetries and exact solutions of the (1+1)-dimensional Toda-like lattice and (2+1) DDE are considered. By setting out the vector field of above two equations, the determine equations will be constructed under the condition of the form invariance of equations. By solving the determine equations, the symmetries of the original equations will be obtained. Similarity reductions of equations are given by solving characteristic equations. Last, the exact solutions are obtained. Furthermore, symmetries and equivalence group transformations of differential-difference equations are also considered.The main contents of this paper are as follows:Chapter one is an introduction. The background of this article, related research, development overview and some basic concepts are introduced. In chapter two,(1+1)-dimensional Toda-like lattice and (2+1)-dimensional DDE equation are studied by using the intrinsic Lie symmetry analysis method. Then similarity reductions and exact solutions are given out.In chapter three, symmetries and equivalence group transformations of differential-difference equations are considered. By means of intrinsic infinitesimal operator method, intrinsic symmetries and equivalence group transformations are given out.The fourth chapter is the summary about the research work of this paper and the further research.