{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/122014"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/122014","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Stability bounds for nonlinear dispersive Hamiltonian partial differential equations","abstract":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-03-01 without embargo terms","abstract_html":"Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-03-01 without embargo terms","abstract_has_math":false,"creators":["Simpson, Sarah E"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Bronski, Jared","DeVille, Lee","Hur, Vera","Rapti, Zoi"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023-12","date_published":"2023-12","updated_at":"2026-07-22T22:25:00Z","subjects":["Partial Differential Equations","Dispersive Equations","Hamiltonian Equations","Gershgorin Disc Theorem","Spectral Stability"],"languages":["en","eng"],"rights":["Copyright 2023 Sarah E. Simpson"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/2142/122014","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bronski, Jared","DeVille, Lee","Hur, Vera","Rapti, Zoi"]},{"key":"dc:creator","label":"Author","values":["Simpson, Sarah E"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2023-12","2023-11-27"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Partial Differential Equations","Dispersive Equations","Hamiltonian Equations","Gershgorin Disc Theorem","Spectral Stability"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en","eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2023 Sarah E. Simpson"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://hdl.handle.net/2142/122014"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-03-01 without embargo terms","The student, Sarah Simpson, accepted the attached license on 2023-11-26 at 19:56.","The student, Sarah Simpson, submitted this Dissertation for approval on 2023-11-26 at 20:18.","This Dissertation was approved for publication on 2023-11-27 at 11:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20006 on 2024-03-01 at 13:14:59","Existing approaches for analyzing the stability of periodic traveling wave solutions to dispersive partial differential equations (PDEs) often rely on numerical methods. However, due to computational limitations, these methods require an implicit assumption that all instabilities lie within a bounded region containing the origin. Recent research has revealed that this assumption does not always hold true, as in certain cases, the spectrum extends to infinity along curves having non-zero real part. Numerical methods are also susceptible to missing spectrum with small real part located further up the imaginary axis. To address these concerns, explicit bounds indicating the region within which all instabilities must be contained would be valuable. This dissertation focuses on providing such bounds for a specific subset of these problems, namely, dispersive, Hamiltonian PDEs. We provide explicit bounds for the region within which all instabilities must lie, along with giving an upper bound on the number of instabilities. For cases where the dispersion relation exhibits at least cubic growth rate these bounds are obtained via a Gershgorin disc theorem type argument coupled with application of the fact that the spectrum of Hamiltonian operators is symmetric with respect to reflection across both the real and imaginary axes. In instances where the growth rate of the dispersion relation is only quadratic we provide an alternative argument following the form of a second-order perturbation calculation to bound the desired region. We provide these results in a general form readily available for applications along with discussing several specific examples in detail including generalized Korteweg-de Vries, Benjamin-Bona-Mahony, Kawahara, and generalized Benjamin-Ono."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Stability bounds for nonlinear dispersive Hamiltonian partial differential equations"]}]}],"canonical_facts":{"dc:contributor":["Bronski, Jared","DeVille, Lee","Hur, Vera","Rapti, Zoi"],"dc:creator":["Simpson, Sarah E"],"dc:date":["2023-12","2023-11-27"],"dc:description":["Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2024-03-01 without embargo terms","The student, Sarah Simpson, accepted the attached license on 2023-11-26 at 19:56.","The student, Sarah Simpson, submitted this Dissertation for approval on 2023-11-26 at 20:18.","This Dissertation was approved for publication on 2023-11-27 at 11:46.","DSpace SAF Submission Ingestion Package generated from Vireo submission #20006 on 2024-03-01 at 13:14:59","Existing approaches for analyzing the stability of periodic traveling wave solutions to dispersive partial differential equations (PDEs) often rely on numerical methods. However, due to computational limitations, these methods require an implicit assumption that all instabilities lie within a bounded region containing the origin. Recent research has revealed that this assumption does not always hold true, as in certain cases, the spectrum extends to infinity along curves having non-zero real part. Numerical methods are also susceptible to missing spectrum with small real part located further up the imaginary axis. To address these concerns, explicit bounds indicating the region within which all instabilities must be contained would be valuable. This dissertation focuses on providing such bounds for a specific subset of these problems, namely, dispersive, Hamiltonian PDEs. We provide explicit bounds for the region within which all instabilities must lie, along with giving an upper bound on the number of instabilities. For cases where the dispersion relation exhibits at least cubic growth rate these bounds are obtained via a Gershgorin disc theorem type argument coupled with application of the fact that the spectrum of Hamiltonian operators is symmetric with respect to reflection across both the real and imaginary axes. In instances where the growth rate of the dispersion relation is only quadratic we provide an alternative argument following the form of a second-order perturbation calculation to bound the desired region. We provide these results in a general form readily available for applications along with discussing several specific examples in detail including generalized Korteweg-de Vries, Benjamin-Bona-Mahony, Kawahara, and generalized Benjamin-Ono."],"dc:format":["application/pdf"],"dc:identifier":["https://hdl.handle.net/2142/122014"],"dc:language":["en","eng"],"dc:rights":["Copyright 2023 Sarah E. Simpson"],"dc:subject":["Partial Differential Equations","Dispersive Equations","Hamiltonian Equations","Gershgorin Disc Theorem","Spectral Stability"],"dc:title":["Stability bounds for nonlinear dispersive Hamiltonian partial differential equations"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:00Z"}