Persistent Link:
http://hdl.handle.net/10150/187122
Title:
Classical and quantum mechanical studies of nonlinear lattices.
Author:
Hays, Mark Hanna, IV.
Issue Date:
1995
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:
A class of nonlinear Hamiltonian lattice models that includes both the nonintegrable discrete nonlinear Schrodinger and the integrable Ablowitz-Ladik models is investigated classically and quantum mechanically. In general, the model under consideration is nonintegrable and its Hamiltonian structure is derived from a nonstandard Poisson bracket. It is shown that solutions of the classical model can, under appropriate and well-defined conditions, become infinite in finite time (blowup). Under suitable restrictions, it is demonstrated that an associated quantum lattice does not exhibit this singular behavior. In this sense, quantum mechanics can regularize a singular classical phenomenon. A fully nonlinear modulation theory for plane wave solutions of the classical lattice is developed. For cases of the model exhibiting blowup, numerical evidence is presented that suggests the existence of both stable and unstable modulated wavetrains. At the present time, it is unclear the extent to which one may relate the onset of instability to blowup. The Hartree approximation is applied to a generalized discrete self-trapping equation (GDST), with the result that the effective Hartree dynamics are described by a rescaled version of the GDST itself. In this manner, the Hartree approximation gives a direct connection between classical and quantum lattice models. Finally, Weyl's ordering prescription is shown to reproduce perturbative results for a weakly nonlinear oscillator. These results are extended to Hamiltonians that are nonpolynomial functions of the number operator. Extensions to the methodology that permit the treatment of other ordering prescriptions are given and compared with Weyl's rule.
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:
Scott, Alwyn

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleClassical and quantum mechanical studies of nonlinear lattices.en_US
dc.creatorHays, Mark Hanna, IV.en_US
dc.contributor.authorHays, Mark Hanna, IV.en_US
dc.date.issued1995en_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.abstractA class of nonlinear Hamiltonian lattice models that includes both the nonintegrable discrete nonlinear Schrodinger and the integrable Ablowitz-Ladik models is investigated classically and quantum mechanically. In general, the model under consideration is nonintegrable and its Hamiltonian structure is derived from a nonstandard Poisson bracket. It is shown that solutions of the classical model can, under appropriate and well-defined conditions, become infinite in finite time (blowup). Under suitable restrictions, it is demonstrated that an associated quantum lattice does not exhibit this singular behavior. In this sense, quantum mechanics can regularize a singular classical phenomenon. A fully nonlinear modulation theory for plane wave solutions of the classical lattice is developed. For cases of the model exhibiting blowup, numerical evidence is presented that suggests the existence of both stable and unstable modulated wavetrains. At the present time, it is unclear the extent to which one may relate the onset of instability to blowup. The Hartree approximation is applied to a generalized discrete self-trapping equation (GDST), with the result that the effective Hartree dynamics are described by a rescaled version of the GDST itself. In this manner, the Hartree approximation gives a direct connection between classical and quantum lattice models. Finally, Weyl's ordering prescription is shown to reproduce perturbative results for a weakly nonlinear oscillator. These results are extended to Hamiltonians that are nonpolynomial functions of the number operator. Extensions to the methodology that permit the treatment of other ordering prescriptions are given and compared with Weyl's rule.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.chairScott, Alwynen_US
dc.contributor.committeememberErcolani, Nicholasen_US
dc.contributor.committeememberWright, Ewan M.en_US
dc.contributor.committeememberFaris, William G.en_US
dc.identifier.proquest9531141en_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.