Two integrable lattice hierarchies and their respective Darboux transformations

Two integrable lattice hierarchies associated with two discrete matrix spectral problems are derived, and the infinitely many conservation laws of the first integrable model is obtained. Moreover, the Darboux transformations based on different Darboux matrixes (2.3.4) and (3.2.4) for the above-mentioned integrable hierarchies are established with the help of two different gauge transformations of lax pairs in this paper. As an application, the explicit solutions of the two integrable hierarchies are presented.