**Habilitation thesis (2012)**

Faculty of Mathematics, University of Vienna.

## Algebro-geometric solutions and their perturbations

### J. Michor

**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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