A Hierarchy of Lattice Soliton Equations Associated with a New Discrete Eigenvalue Problem and Darboux Transformations

By considering a new discrete isospectral eigen-value problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.