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dc.contributor.advisorPrinari, Barbara
dc.contributor.authorOrtiz, Alyssa Kayelin
dc.contributor.committeememberCascaval, Radu
dc.contributor.committeememberMorrow, Greg
dc.contributor.committeememberBihun, Oksana
dc.contributor.committeememberGrabowski, Marek
dc.date.accessioned2019-05-29T13:40:49Z
dc.date.available2019-05-29T13:40:49Z
dc.date.submitted2019-05
dc.descriptionIncludes bibliographical references.
dc.description.abstractThe scalar nonlinear Schrodinger (NLS) equation is a nonlinear partial differential equation that models many of the world’s weakly nonlinear dispersive wave phenomena. This dissertation discusses two important variants of NLS: (i) the matrix NLS (MNLS), applicable in low-temperature physics and nonlinear optics, and (ii) the defocusing Ablowitz-Ladik equation, a variant of scalar NLS, continuous in time but discrete in space. In Chapter 2, the Inverse Scattering Transform (IST) is used to investigate four distinct MNLS systems, proposed by Tsuchida, shown to be integrable and to have solitary wave solutions (i.e. solitons). Two of these systems correspond to focusing and defocusing MNLS equations; the other two exhibit mixed-sign Minkowski-type nonlinear terms. In this chapter, the IST is developed with zero boundary conditions (ZBC) for all four systems, completely characterizing the solution’s spectrum. Novel soliton solutions are also derived for the mixed-sign MNLS systems. In Chapter 3, these MNLS systems are analyzed with nonzero boundary conditions (NZBC) using IST. The NZBC introduce challenges due to the existence of branch points in the spectral parameter, but the reward is a richer set of soliton solutions, such as dark solitons, periodic solutions, and rational solutions (i.e., rogue waves). In Chapter 4, the IST is developed for the defocusing Ablowitz-Ladik (AL) equation with an arbitrarily large nonzero background. It is well known that continuous and discrete focusing NLS systems exhibit modulational instability, i.e., the instability of a constant background with respect to long wavelength perturbations. In the continuous scalar defocusing NLS, modulational instability is never observed, regardless of background size. However, Ohta and Yang recently showed that the defocusing AL equation with a background greater than one becomes modulationally unstable, admitting rational solutions that are the analog of those found in its focusing counterpart. These recent findings motivate our investigation since the IST for the defocusing AL equation with NZBC has only previously been developed under the assumption of a small background less than one. The results presented in Chapter 2 were published in Studies in Applied Mathematics, and those in Chapters 3 and 4 have been submitted for publication.
dc.identifierOrtiz_uccs_0892D_10479.pdf
dc.identifier.urihttps://hdl.handle.net/10976/167119
dc.languageEnglish
dc.publisherUniversity of Colorado Colorado Springs. Kraemer Family Library
dc.relation.ispartofDissertations
dc.rightsCopyright of the original work is retained by the author.
dc.subjectInverse scattering transform
dc.subjectSoliton solution
dc.subjectNonlinear schrodinger equation
dc.subjectAblowitz-ladik equation
dc.titleINVERSE SCATTERING TRANSFORM AND SOLITONS FOR MATRIX NONLINEAR SCHRODINGER SYSTEMS AND FOR THE DEFOCUSING ABLOWITZ-LADIK EQUATION WITH NONZERO BOUNDARY CONDITIONS
dc.typeText
dcterms.cdm.subcollectionMathematics
thesis.degree.disciplineCollege of Letters, Arts, and Sciences-Mathematics
thesis.degree.grantorUniversity of Colorado Colorado Springs
thesis.degree.levelDoctoral
thesis.degree.nameDoctor of Philosophy (Ph.D.)


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