Date of Award:

5-2014

Document Type:

Dissertation

Degree Name:

Doctor of Philosophy (PhD)

Department:

Mathematics and Statistics

Committee Chair(s)

Zhi-Qiang Wang

Committee

Zhi-Qiang Wang

Committee

Ian Anderson

Committee

Luis Gordillo

Committee

Joe Koebbe

Committee

Lie Zhu

Abstract

We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations:

I) (Special) Relativistic Pendulum Equations (RPEs);

II) (2-coupled) Gross-Pitaevskii Equations (GPEs).

The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will flow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation is called RPE. Our result answers some open questions about the existence of multiple periodic solutions for RPEs.

GPEs are sometimes called coupled nonlinear Schrödinger equations. The Schrödinger equation is the fundamental equation of Quantum Mechanics which is the "exotic" probabilistic fundamental physics law of the "micro" world — the world of atoms and molecules. A well-known physicist and Nobel laureate, R. Feynman, said "I think I can safely say that nobody understands quantum mechanics." which indicates the physical/ philosophical difficulty of interpretations. It raises paradoxical problems such the well-known Schrödinger's Cat. Setting aside these difficulties, if we combine Special Relativity and Quantum Mechanics as a many-body system, then we have Quantum Field Theory (QFT) which is more deterministic, and governs even elementary particle physics. GPEs are also related to QFT. For example, superconductivity and Bose Einstein Condensates (BEC). These phenomena in condensed matter physics can be thought of as the emergence of the mysterious micro world physics at "macro" level.

We study these equations from the viewpoint of mathematical interest. It is generally difficult to solve nonlinear differential equations. It is also generally difficult even to prove the existence of solutions. Although we show there exist solutions, we still do not know how to solve the differential equations analytically.

Variational Methods (or Calculus of Variations) are useful tools to show there exist solutions of differential equations. The idea is to convert the problem of solving equations into the problem of finding critical points (i.e. minimum/maximum points or saddle points) of a functional, and each critical point can generally correspond to a weak solution. However, it is also generally difficult to find out such critical points because we look for critical points in an infinite-dimensional functions space. Thus many advanced mathematical theories or tools have been developed and used for decades in nonlinear analysis. We use some topological theories. From information of the functional's shape, these theories deduce if there exists a critical point, or how many critical points exist. The key of these theories is to use the symmetry of the equations.

We also investigate bifurcation structures for II), i.e. the connection structures between the solutions. By linearizations which look at the equations "locally," we reduce the problem in the infinite dimension to one in a finite dimension. Furthermore, it allows us to apply Morse Theory, which connects between local and global aspects of the functional's information. In several cases, we show that there are infinitely many bifurcation points that give rise to global bifurcation branches.

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