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Utah State University

1. Tian, Rushun. Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations.

Degree: PhD, Mathematics and Statistics, 2013, Utah State University

URL: https://digitalcommons.usu.edu/etd/1484

Coupled nonlinear Schrodinger equations (CNLS) govern many physical phenomena, such as nonlinear optics and Bose-Einstein condensates. For their wide applications, many studies have been carried out by physicists, mathematicians and engineers from different respects. In this dissertation, we focused on standing wave solutions, which are of particular interests for their relatively simple form and the important roles they play in studying other wave solutions. We studied the multiplicity of this type of solutions of CNLS via variational methods and bifurcation methods.
Variational methods are useful tools for studying differential equations and systems of differential equations that possess the so-called variational structure. For such an equation or system, a weak solution can be found through finding the critical point of a corresponding energy functional. If this equation or system is also invariant under a certain symmetric group, multiple solutions are often expected. In this work, an integer-valued function that measures symmetries of CNLS was used to determine critical values. Besides variational methods, bifurcation methods may also be used to find solutions of a differential equation or system, if some trivial solution branch exists and the system is degenerate somewhere on this branch. If local bifurcations exist, then new solutions can be found in a neighborhood of each bifurcation point. If global bifurcation branches exist, then there is a continuous solution branch emanating from each bifurcation point.
We consider two types of CNLS. First, for a fully symmetric system, we introduce a new index and use it to construct a sequence of critical energy levels. Using variational methods and the symmetric structure, we prove that there is at least one solution on each one of these critical energy levels. Second, we study the bifurcation phenomena of a two-equation asymmetric system. All these bifurcations take place with respect to a positive solution branch that is already known. The locations of the bifurcation points are determined through an equation of a coupling parameter. A few nonexistence results of positive solutions are also given
*Advisors/Committee Members: Zhi-Qiang Wang, ;.*

Subjects/Keywords: Bifurcation; Coupled nonlinear Schrodinger equations; Indefinite; Standing wave solutions; Z_N symmetry; Physical Sciences and Mathematics; Statistics and Probability

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APA (6^{th} Edition):

Tian, R. (2013). Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations. (Doctoral Dissertation). Utah State University. Retrieved from https://digitalcommons.usu.edu/etd/1484

Chicago Manual of Style (16^{th} Edition):

Tian, Rushun. “Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations.” 2013. Doctoral Dissertation, Utah State University. Accessed September 26, 2020. https://digitalcommons.usu.edu/etd/1484.

MLA Handbook (7^{th} Edition):

Tian, Rushun. “Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations.” 2013. Web. 26 Sep 2020.

Vancouver:

Tian R. Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations. [Internet] [Doctoral dissertation]. Utah State University; 2013. [cited 2020 Sep 26]. Available from: https://digitalcommons.usu.edu/etd/1484.

Council of Science Editors:

Tian R. Existence and Multiplicity Results on Standing Wave Solutions of Some Coupled Nonlinear Schrodinger Equations. [Doctoral Dissertation]. Utah State University; 2013. Available from: https://digitalcommons.usu.edu/etd/1484