Persistent Link:
http://hdl.handle.net/10150/186987
Title:
Integrable curve dynamics.
Author:
Calini, Annalisa Maria.
Issue Date:
1994
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:
The Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³. The role of orthonormal frames of the curve is explored. In this framework a second Poisson structure for the Heisenberg Model is derived and the relation between the Heisenberg Model and the cubic Non-Linear Schrodinger Equation is explained. The Frenet frame of a curve is shown to be a Legendrian curve in the space of orthonormal frames with respect to a natural contact structure. As a consequence, generic singularities of the solution of the Heisenberg Model and topological invariants of the curve are computed. The family of multi-phase solutions of the Heisenberg Model and the corresponding curves are constructed with techniques of algebraic geometry. The relation with the Non-Linear Schrodinger Equation is explained also in this context. A formula for the Backlund transformation for the Heisenberg Model is derived and applied to construct orbits homoclinic to planar circles. As a result singular knots are obtained.
Type:
text; Dissertation-Reproduction (electronic)
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Committee Chair:
Ercolani, Nicholas

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleIntegrable curve dynamics.en_US
dc.creatorCalini, Annalisa Maria.en_US
dc.contributor.authorCalini, Annalisa Maria.en_US
dc.date.issued1994en_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.abstractThe Heisenberg Model of the integrable evolution of a continuous spin chain can be used to describe an integrable dynamics of curves in R ³. The role of orthonormal frames of the curve is explored. In this framework a second Poisson structure for the Heisenberg Model is derived and the relation between the Heisenberg Model and the cubic Non-Linear Schrodinger Equation is explained. The Frenet frame of a curve is shown to be a Legendrian curve in the space of orthonormal frames with respect to a natural contact structure. As a consequence, generic singularities of the solution of the Heisenberg Model and topological invariants of the curve are computed. The family of multi-phase solutions of the Heisenberg Model and the corresponding curves are constructed with techniques of algebraic geometry. The relation with the Non-Linear Schrodinger Equation is explained also in this context. A formula for the Backlund transformation for the Heisenberg Model is derived and applied to construct orbits homoclinic to planar circles. As a result singular knots are obtained.en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.chairErcolani, Nicholasen_US
dc.contributor.committeememberFlaschka, Hermannen_US
dc.contributor.committeememberPickrell, Dougen_US
dc.identifier.proquest9517597en_US
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