Alexandria Digital Research Library

Semiclassical methods for high frequency wave propagation in periodic media

Author:
Delgadillo, Ricardo A.
Degree Grantor:
University of California, Santa Barbara. Mathematics
Degree Supervisor:
Xu Yang
Place of Publication:
[Santa Barbara, Calif.]
Publisher:
University of California, Santa Barbara
Creation Date:
2016
Issued Date:
2016
Topics:
Quantum physics and Mathematics
Keywords:
Wave
Semiclassical
Propagation
High
Frequency
Periodic
Genres:
Dissertations, Academic and Online resources
Dissertation:
Ph.D.--University of California, Santa Barbara, 2016
Description:

We will study high-frequency wave propagation in periodic media. A typical example is given by the Schrodinger equation in the semiclassical regime with a highly oscillatory periodic potential and external smooth potential. This problem presents a numerical challenge when in the semiclassical regime. For example, conventional methods such as finite differences and spectral methods leads to high numerical cost, especially in higher dimensions. For this reason, asymptotic methods like the frozen Gaussian approximation (FGA) was developed to provide an efficient computational tool. Prior to the development of the FGA, the geometric optics and Gaussian beam methods provided an alternative asymptotic approach to solving the Schrodinger equation efficiently. Unlike the geometric optics and Gaussian beam methods, the FGA does not lose accuracy due to caustics or beam spreading.

In this thesis, we will briefly review the geometric optics, Gaussian beam, and FGA methods. The mathematical techniques used by these methods will aid us in formulating the Bloch-decomposition based FGA. The Bloch-decomposition FGA generalizes the FGA to wave propagation in periodic media. We will establish the convergence of the Bloch-decomposition based FGA to the true solution for Schrodinger equation and develop a gauge-invariant algorithm for the Bloch-decomposition based FGA. This algorithm will avoid the numerical difficulty of computing the gauge-dependent Berry phase. We will show the numerical performance of our algorithm by several one-dimensional examples.

Lastly, we will propose a time-splitting FGA-based artificial boundary conditions for solving the one-dimensional nonlinear Schrodinger equation (NLS) on an unbounded domain. The NLS will be split into two parts, the linear and nonlinear parts. For the linear part we will use the following absorbing boundary strategy: eliminate Gaussian functions whose centers are too distant to a fixed domain.

Physical Description:
1 online resource (96 pages)
Format:
Text
Collection(s):
UCSB electronic theses and dissertations
ARK:
ark:/48907/f38c9w98
ISBN:
9781369146974
Catalog System Number:
990046968240203776
Rights:
Inc.icon only.dark In Copyright
Copyright Holder:
Ricardo Delgadillo
File Description
Access: Public access
Delgadillo_ucsb_0035D_13047.pdf pdf (Portable Document Format)