Solving the non-isospectral Ablowitz-Ladik hierarchy via the inverse scattering transform and reductions

The non-isospectral Ablowitz-Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz-Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz-Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.