Darboux Transformations Of Two Nonlinear Differential-Difference Equations

In this paper, we shall study two differential-difference equations associated with discrete 3×3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformations of the differential-difference equations are given. In order to solve the two differential-difference equations, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equations are given.