Periodic processes and dispersion relations for Ablowitz–Ladik equations

There is considered the system of algebraic dispersion equations for the complexified Ablowitz–Ladik discrete equations and relevant reductions to “dark” and “white” versions, in an analogy to the NLS equation. An universal formalism presented here allows to determine effectively parameters of multiphase quasiperiodic solutions, multisoliton ones and of solutions in a form of solitons on a quasiperiodic background. This idea is connected with so-called soliton limit, which in the case of the white type solution, differs essentially from those introduced in the past and must be preceded by some modular transformation. There is a full correspondence with a similar approach for NLS equations. A few subtleties concerning solvability and examples are also discussed.