Darboux Transformation Of Two Differential-difference Equations Associated With A Discrete 3×3 Matrix Spectral Problem

In this paper, we mainly study the Darboux transformation of two differential-difference equations, which related to the same discrete 3×3 matrix spectral problem. Beginning from constructing a gauge transformation of the discrete 3×3 matrix spectral problem, we prove that the Lax pairs of the two differential-difference equations both keep their form invariant under the gauge transformation, from where we get the Darboux transformation of the two differential-difference equations. As an application, we obtain the explicit solutions of the two differential-difference equations by the formerly constructed Darboux transformation and their graph.