Lax Pairs for the Ablowitz-Ladik System Via Orthogonal Polynomials on the Unit Circle

Author: Nenciu, Irina

Year: 2005

Degree: Dissertation (Ph.D.)

Advisor: Simon, Barry M.

Committee Members: Simon, Barry M.; Damanik, David; Killip, Rowan; Kaloshin, Vadim

Option: Mathematics

DOI: 10.7907/462M-V013

Abstract

We investigate the existence and properties of an integrable system related to orthogonal polynomials on the unit circle. We prove that the main evolution of the system is defocusing Ablowitz-Ladik (also known as the integrable discrete nonlinear Schroedinger equation). In particular, we give a new proof of complete integrability for this system.

Furthermore, we use the CMV and extended CMV matrices defined in the context of orthogonal polynomials on the unit circle by Cantero, Moral, and Velazquez, and Simon, respectively, to construct Lax pair representations for the Ablowitz-Ladik hierarchy in the periodic, finite, and infinite settings.

Files

• Thesis.pdf (application/pdf)