An investigation of the Novikov-Veselov equation: new solutions, stability and implications for the inverse scattering transform
| dc.contributor.author | Croke, Ryan P., author | |
| dc.contributor.author | Mueller, Jennifer, advisor | |
| dc.contributor.author | Bradley, Mark, committee member | |
| dc.contributor.author | Shipman, Patrick, committee member | |
| dc.contributor.author | Zhou, Yongcheng, committee member | |
| dc.date.accessioned | 2007-01-03T08:09:45Z | |
| dc.date.available | 2007-01-03T08:09:45Z | |
| dc.date.issued | 2012 | |
| dc.description.abstract | Integrable systems in two spatial dimensions have received far less attention by scholars than their one--dimensional counterparts. In this dissertation the Novikov--Veselov (NV) equation, a (2+1)--dimensional integrable system that is a generalization of the famous Korteweg de--Vreis (KdV) equation is investigated. New traveling wave solutions to the NV equation are presented along with an analysis of the stability of certain types of soliton solutions to transverse perturbations. To facilitate the investigation of the qualitative nature of various types of solutions, including solitons and their stability under transverse perturbations, a version of a pseudo-spectral numerical method introduced by Feng [J. Comput. Phys., 153(2), 1999] is developed. With this fast numerical solver some conjectures related to the inverse scattering method for the NV equation are also examined. The scattering transform for the NV equation is the same as the scattering transform used to solve the inverse conductivity problem, a problem useful in medical applications and seismic imaging. However, recent developments have shed light on the nature of the long-term behavior of certain types of solutions to the NV equation that cannot be investigated using the inverse scattering method. The numerical method developed here is used to research these exciting new developments. | |
| dc.format.medium | born digital | |
| dc.format.medium | doctoral dissertations | |
| dc.identifier | Croke_colostate_0053A_11099.pdf | |
| dc.identifier | ETDF2012400230MATH | |
| dc.identifier.uri | http://hdl.handle.net/10217/67426 | |
| dc.language | English | |
| dc.language.iso | eng | |
| dc.publisher | Colorado State University. Libraries | |
| dc.relation.ispartof | 2000-2019 | |
| dc.rights | Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright. | |
| dc.title | An investigation of the Novikov-Veselov equation: new solutions, stability and implications for the inverse scattering transform | |
| dc.type | Text | |
| dcterms.rights.dpla | This Item is protected by copyright and/or related rights (https://rightsstatements.org/vocab/InC/1.0/). You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s). | |
| thesis.degree.discipline | Mathematics | |
| thesis.degree.grantor | Colorado State University | |
| thesis.degree.level | Doctoral | |
| thesis.degree.name | Doctor of Philosophy (Ph.D.) |
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