We provide a detailed recursive construction of the Ablowitz-Ladik (AL) hierarchy and its zero-curvature formalism. The two-coefficient AL hierarchy under investigation can be considered a complexified version of the discrete nonlinear
Schrödinger equation and its hierarchy of nonlinear evolution equations.
Specifically, we discuss in detail the stationary Ablowitz-Ladik formalism in connection with the underlying hyperelliptic curve and the stationary Baker-Akhiezer function and separately the corresponding time-dependent Ablowitz-Ladik formalism.
MSC2000: Primary 37K15, 37K10; Secondary 39A12, 35Q55
Keywords: Ablowitz-Ladik hierarchy, discrete NLS