Mathematical Theory and Numerical Methods for Gross-Pitaevskii Equations and Applications

Ph.DDOCTOR OF PHILOSOPH

Quantum Analogues of Two-Dimensional Classical Turbulence

Turbulence, the irregular motion of fluids, is a challenging problem in physics. Yet some properties of turbulence appear to be universal, independent of the underlying host fluid supporting the motion. Recent studies have found that turbulence in superfluid helium, a quantum fluid, exhibits two of the most fundamental laws of classical fluid turbulence: the Kolmogorov −5/3 law, and the dissipation anomaly. These laws appear despite the fluid being highly constrained by quantum mechanical effects, and completely lacking kinematic viscosity. Such findings suggest further insight into the universal features of turbulent phenomena can be gained by studying analogies between classical and quantum turbulence. Atomic Bose-Einstein condensates (BECs) offer a new platform for the study of quantum turbulence; the geometric control available in BEC experiments offers the possibility of studying quantum turbulence in effectively two-dimensional fluids. As two-dimensional turbulence exhibits dramatically different features from its 3D counterpart, BEC systems allow for further study of the analogies between classical and quantum turbulence. In this thesis we numerically and theoretically study 2D quantum turbulence in BECs within the framework of the Gross-Pitaevskii model. We focus on analogies with classical 2D turbulence, with the aim of identifying common or universal features. First we investigate coherent vortex structures in negative temperature equilibria via an experimentally accessible flow-field measure. Coherent vortices are shown to produce a clear signal in this measure that is independent of the confinement geometry, and we demonstrate that it can be observed in dynamical simulations. Second, studying a quantum analogue of the two-dimensional cylinder wake, we investigate the phenomenon of Strouhal oscillations. We find that the Strouhal number obeys a universal relation, similar to the classical form, upon introducing a modified superfluid Reynolds number that accounts for the critical velocity for vortex nucleation. Like the classical Reynolds number, the superfluid Reynolds number is found to govern the transition from laminar to turbulent behaviour in the quantum fluid. Finally, simulating decaying 2D quantum turbulence for very large vortex numbers, we show that quantum fluids are capable of supporting the direct enstrophy cascade, a fundamental feature of two-dimensional turbulent flows. The quantum fluid manifests key features of the classical cascade, including Batchelor’s −3 law of the inertial range, scaling of the inertial range against the superfluid Reynolds number, and the value of the Kraichnan-Batchelor constant. The findings from this work thus provide some new insight into the universality of fundamental turbulence concepts, and their applicability to quantum fluids

NASA thesaurus. Volume 1: Hierarchical Listing

There are over 17,000 postable terms and nearly 4,000 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary and Volume 3 - Definitions

NASA Thesaurus. Volume 1: Hierarchical listing

There are 16,713 postable terms and 3,716 nonpostable terms approved for use in the NASA scientific and technical information system in the Hierarchical Listing of the NASA Thesaurus. The generic structure is presented for many terms. The broader term and narrower term relationships are shown in an indented fashion that illustrates the generic structure better than the more widely used BT and NT listings. Related terms are generously applied, thus enhancing the usefulness of the Hierarchical Listing. Greater access to the Hierarchical Listing may be achieved with the collateral use of Volume 2 - Access Vocabulary

NASA Thesaurus. Volume 1: Alphabetical listing

The NASA Thesaurus -- Volume 1, Alphabetical Listing -- contains all subject terms (postable and nonpostable) approved for use in the NASA scientific and technical information system. Included are the subject terms of the Preliminary Edition of the NASA Thesaurus (NASA SP-7030, December 1967); of the NASA Thesaurus Alphabetical Update (NASA SP-7040, September 1971); and terms approved, added or changed through May 31, 1975. Thesaurus structuring, including scope notes, a generic structure with broader-term/narrower-term (BT-NT) relationships displayed in embedded hierarchies, and other cross references, is provided for each term, as appropriate

A Two-Dimensional Dirac System and Envelope Solutions to the Nonlinear Schrödinger Equation

In studying the cubic nonlinear Schrödinger (NLS) equation with hexagonal lattice potential, Ablowitz, Nixon, and Zhu [1] and Fefferman and Weinstein [31] have used ansatz solutions of a periodic, cubic NLS equation to derive two similar two-dimensional Dirac equations with cubic nonlinearity. Chapters 1 and 2 of this thesis deal with solutions and lifespans of solutions for the linear and nonlinear Dirac equations. We establish local and almost global existence results as well as ill-posedness below the critical regularity, H ̇ 1/2. We leave as an open question whether solutions blow up in finite time or if a global existence result can be found. The third chapter modifies the machinery of [32] and [31] to explore an open question posed by Fefferman and Weinstein, [31]. We prove that an envelope of solutions to the slowly modulated Dirac equation provides a good approximation for solutions to the nonlinear Schrödinger equation with hexagonal lattice. The NLS solution is shown to exist for long times with the nonlinear Dirac dynamics affecting the solution on any constant timescale. The same timescale is also proven for an ansatz envelope proposed by Ablowitz, Nixon, and Zhu. The timescale is also extended in an intermediate regime by slightly weakening the nonlinearity of the governing Dirac equation. The final chapter discusses future work focusing on some numerical simulations for the Dirac and Schrödinger equations.Doctor of Philosoph

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described