Family Of Nonlinear Differential Difference Equations And Its Darboux Transformation

A heirachy of nonlinear differential-differenceequations assoiated with a discrete spectral problem areproposed, It is shown that the heirachy of differential-difference equations possesses the hamilton structures.Darboux transformations for the first two differential-difference equations are found,also is that for the (2+1)dimension differential-difference equation. As anapplication, several examples are given.