Dissertations
https://hdl.handle.net/10976/167632
2021-04-21T08:17:42ZLower Finite Modules Over Commutative Rings With Identity
https://hdl.handle.net/10976/167279
Lower Finite Modules Over Commutative Rings With Identity
Harmon, Luke Everett
A bounded partially ordered set (P, 0, 1, ≤) is lower finite providedP is infinite and for each x 6= 1 in P, there are but finitely many elements y in Psuch that y < x. We will call a module M lower finite if the set of proper submodulesof M, partially ordered by set-theoretic containment, is lower finite. We will use the(well-studied) class of Jonsson modules (along with other classical results) to classifythe lower finite modules over a commutative ring with identity.
Includes bibliographical references.
INVERSE SCATTERING TRANSFORM AND SOLITONS FOR MATRIX NONLINEAR SCHRODINGER SYSTEMS AND FOR THE DEFOCUSING ABLOWITZ-LADIK EQUATION WITH NONZERO BOUNDARY CONDITIONS
https://hdl.handle.net/10976/167119
INVERSE SCATTERING TRANSFORM AND SOLITONS FOR MATRIX NONLINEAR SCHRODINGER SYSTEMS AND FOR THE DEFOCUSING ABLOWITZ-LADIK EQUATION WITH NONZERO BOUNDARY CONDITIONS
Ortiz, Alyssa Kayelin
The 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.
Includes bibliographical references.