New hierarchies of integrable positive and negative lattice models and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations associated with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. It is also shown that each equation in the resulting hierarchies is Liouville integrable. As the typical example, a fairly new and infrequent rational equation with regard to potentials is presented. Furthermore, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.