Keywords: Algebro-geometric solutions, Ablowitz-Ladik hierarchy, discrete NLS, Hamiltonian formalism, conservation laws, initial value problem, complex-valued solutions, spatial asymptotics, Toda hierarchy, Jacobi operators, scattering theory, inverse scattering transform, steplike, solitons.

Abstract: The unifying theme of the work presented here are algebro-geometric solutions of hierarchies of nonlinear integrable differential-difference equations continuous in time and discrete in space. Algebro-geometric solutions are a natural extension of the class of soliton solutions and can be explicitly constructed using elements of algebraic geometry. The construction of such solutions in terms of specific algebro-geometric data on a compact hyperelliptic Riemann surface is exemplified for one model, the Ablowitz-Ladik hierarchy. Scattering theory with respect to (two different) algebro-geometric background operators and its application to the inverse scattering transform are studied for a second discrete model, the celebrated Toda hierarchy.

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