Poisson geometry of the Ablowitz-Ladik equations

• Main Author: Lozano, Guadalupe I.
• Other Authors: Ercolani, Nicolas M.
• Language: en_US
• Published: The University of Arizona. 2004
• Subjects: Mathematics.
• Online Access: http://hdl.handle.net/10150/290120

This research seeks to understand the Poisson Geometry of the Ablowitz-Ladik equations (AL), an integrable discretization of the Non-linear Schrodinger equation (NLS) first proposed by Ablowitz and Ladik in the 70's. More specifically, to argue that the AL hierarchy (an integrable hierarchy of equations which comprises AL) can be explicitly viewed as a hierarchy of commuting flows which: (1) are Hamiltonian with respect to both a (known) Poisson operator J, and a (new) non-local, skew, almost Poisson operator K, on the appropriate space; (2) can be recursively generated from an operator R = KJ⁻¹. This thesis also clarifies the geometric framework that underlies a certain class of evolving geodesic linkages related to the AL hierarchy via the evolution for their "discrete" geodesic curvature. In this regard, our results include a geometric interpretation of a compatibility condition associated to a Lax pair for AL and, consequently, a bijective correspondence between AL flows and linkage flows.