Persistent Link:
http://hdl.handle.net/10150/290120
Title:
Poisson geometry of the Ablowitz-Ladik equations
Author:
Lozano, Guadalupe I.
Issue Date:
2004
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
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.
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
Ercolani, Nicolas M.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titlePoisson geometry of the Ablowitz-Ladik equationsen_US
dc.creatorLozano, Guadalupe I.en_US
dc.contributor.authorLozano, Guadalupe I.en_US
dc.date.issued2004en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractThis 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.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorErcolani, Nicolas M.en_US
dc.identifier.proquest3145093en_US
dc.identifier.bibrecord.b47210904en_US
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