Abstract. We will study 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 similar to these, they 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 will be exemplified for one model, the Ablowitz-Ladik hierarchy, which is a complexified version of the discrete nonlinear Schroedinger hierarchy. We derive Riemann theta function representations for the algebro-geometric solutions and present a new algorithm to solve the inverse algebro-geometric spectral problem for general Ablowitz-Ladik Lax operators, starting from initial divisors in a dense set of full measure.

Perturbations of algebro-geometric solutions, or more precisely, scattering theory with respect to (two different) algebro-geometric background operators and its application to the inverse scattering transform will be discussed for a second discrete model, the Toda hierarchy, if time permits.

Von 15:45 - 16:15 Uhr, Kaffeejause im common room.

Die offizielle Einladung findet sich hier.