On the Ablowitz-Ladik equations with self-consistent sources

Darboux transformations and explicit solutions to Ablowitz-Ladik (AL) equations with self-consistent sources (ALESCS) are studied. Based on the Darboux transformation (DT) for the AL problem, we construct three types of non-auto-Baecklund transformations connected with AL systems with different numbers of sources. The degenerate cases of DT and their applications to the reduced systems of ALESCS, for instance, discrete nonlinear Schroedinger with self-consistent sources (D-NLSSCS) and discrete mKdV equation with self-consistent sources (D-mKdVSCS), are discussed. Many types of solutions of ALESCS, D-NLSSCS and D-mKdVSCS, including solitons, positons, negatons can be derived from DTs and their degenerate cases