Long-Time Asymptotics for the Toda Shock Problem: Non-Overlapping Spectra

We derive the long-time asymptotics for the Toda shock problem using the nonlinear steepest descent analysis for oscillatory Riemann--Hilbert factorization problems. We show that the half plane of space/time variables splits into five main regions: The two regions far outside where the solution is close to free backgrounds. The middle region, where the solution can be asymptotically described by a two band solution, and two regions separating them, where the solution is asymptotically given by a slowly modulated two band solution. In particular, the form of this solution in the separating regions verifies a conjecture from Venakides, Deift, and Oba from 1991.Comment: 39 page

Scattering Theory for Jacobi Operators with Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely.Comment: 29 page

Scattering Theory for Jacobi Operators with General Steplike Quasi-Periodic Background

We develop direct and inverse scattering theory for Jacobi operators with steplike coefficients which are asymptotically close to different finite-gap quasi-periodic coefficients on different sides. We give a complete characterization of the scattering data, which allow unique solvability of the inverse scattering problem in the class of perturbations with finite first moment.Comment: 23 page

Algebro-Geometric Finite-Gap Solutions of the Ablowitz-Ladik Hierarchy

We provide a detailed derivation of all complex-valued algebro-geometric finite-band solutions of the Ablowitz-Ladik hierarchy. In addition, we survey a recursive construction of the Ablowitz-Ladik hierarchy and its zero-curvature and Lax formalism.Comment: 41 page

Trace formulas and inverse spectral theory for finite Jacobi operators

Diese Arbeit untersucht folgende Problemstellung: welche Spektraldaten eines endlichen Jacobioperators reichen aus, um den Operator eindeutig zu rekonstruieren. Wir beweisen, dass N Eigenwerte einer N × N Jacobi Matrix J zusammen mit N − 1 Eigenwerten von zwei Teilmatrizen die Jacobi Matrix eindeutig bestimmen. Die Teilmatrizen erhalten wir durch Streichen der n-ten Zeile und Spalte von J. Hinreichende und notwendige Bedingungen an die Eigenwerte werden gegeben, aus denen die Existenz einer zugehoerigen Jacobi Matrix folgt. In der Physik beschreibt dieses Modell eine Kette von N Massenpunkten mit fixierten Enden, die durch Federn miteinander verbunden sind. Aus den Eigenfrequenzen dieses Systems und des Systems, in dem ein weiterer innerer Punkt festgehalten wird, koennen die Massen und die Federkonstanten des urspruenglichen Systems eindeutig rekonstruiert werden.The goal of this thesis is to determine spectral data of finite Jacobi operators which are necessary and sufficient to reconstruct the operator uniquely. We prove that N eigenvalues of a N × N Jacobi matrix J together with N −1 eigenvalues of two submatrices of J which we obtain by omitting the n-th line and column uniquely determine J. Necessary and sufficient restrictions on the sets of eigenvalues are given under which one obtains existence of J. From a physical point of view such a model describes a chain of N particles coupled via springs and fixed at both end points. Determining the eigenfrequencies of this system and the one obtained by keeping one particle fixed, one can uniquely reconstruct the masses and spring constants

Scattering theory for Jacobi operators and applications to completely integrable systems

In der vorliegenden Arbeit wird die direkte und inverse Streutheorie fuer Jacobioperatoren entwickelt, die kurzreichweitige Perturbationen von quasi-periodischen finite-gap Operatoren sind. Wir zeigen Existenz des Transformationsoperators, untersuchen dessen Eigenschaften, leiten die Gel'fand-Levitan-Marchenko Gleichung her und geben minimale Streudaten an, die den gestoerten Operator eindeutig beschreiben. Weiters wird das zugehoerige Anfangswertproblem der Todahierachie mittels der inversen Streutransformation geloest.In this thesis we develop direct and inverse scattering theory for Jacobi operators which are short range perturbations of quasi-periodic finite-gap operators. We show existence of transformation operators, investigate their properties, derive the corresponding Gel'fand-Levitan-Marchenko equation, and find minimal scattering data which determine the perturbed operator uniquely. Then we apply this knowledge to solve the associated initial value problem of the Toda hierarchy via the inverse scattering transform

Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds

We provide a rigorous treatment of the inverse scattering transform for the entire Toda hierarchy for solutions which are asymptotically close to (in general) different finite-gap solutions as $n\to\pm\infty$.Comment: 10 page