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 National Science
Foundation Award #0505681 |
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Mathematical Analysis for Problems in Nonlinear Optics |
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| Investigator(s): |
Jamison Moeser (PI)
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| Sponsor: |
University of Colorado at Boulder, CO 80309 3034926221
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| Start Date/Expiration Date |
2005-06-15 to 2008-05-31 (amended 2005-06-14) |
| Awarded Amount to Date: |
$77,642 |
| Abstract: Abstract, DMS-0505681, J Moeser, University of Colorado
Title: Mathematical Analysis for Problems in Nonlinear Optics
The purpose of this project is to develop and implement
analytical techniques for the study of problems arising in nonlinear
optics. The array of applications ranges from light pulse propagation in
optical fibers and nonlinear waveguides to stabilization of
spatio-temporal solitons, both for deterministic systems and systems
with a stochastic component. Depending on the particular application,
the relevant mathematical models are nonlinear partial
differential equations, integral equations, or discrete equations.
The core mathematical issues are fundamental
understanding of asymptotic theories, both analytically and
numerically, well posedness or blow up for the modeling equations, and
the search for solitary wave solutions via direct methods of calculus
of variations. This project focuses on three major applications in
nonlinear optics: systems with randomness, discrete systems, and
generalized dispersion managed systems. Randomness arises
ubiquitously in nonlinear optics and an important goal of this project
is to study noise models that are physically relevant and analyzable
in a rigorous probabilistic sense. Such models have already yielded
new types of fundamental solitons and their study may help engineers
to design optical devices whose performance is actually enhanced by
small amounts of randomness. Discrete equations arise in many
important contexts, including the study of nonlinear waveguides,
all-optical switches, and Bose-Einstein condensates. Discrete models
have very different properties than those of their continuous
counterparts, and though regularity of solutions in the spatial
variable is no longer an issue, the lack of scaling invariances in
general makes their analysis more difficult. Thus discrete problems
invite the development of new analytical tools which ultimately will
help explain experimental observations. Finally, the P.I. will study
generalized dispersion managed systems. The technique of dispersion
management has been instrumental in enabling higher bit rate
communications and has provided the impetus for many interesting
mathematical investigations. The approximate models that arise have a
nonlocal nonlinearity which presents interesting mathematical and
numerical challenges. Often the nonlocality in the modeling equation
reflects a stabilizing effect in the original physical system, and an
important recurring question is how to exploit the nonlocal structure
in order to help explain the observed stabilization. The P.I. will
examine the application of dispersion management technology in
contexts other than fiber optic communications, such as the search for
spatio-temporal solitons, where the P.I. expects to obtain new
information on it's possible stabilizing effects.
Fundamental mathematical understanding of nonlinear optical
systems is central to the development of technologies that will be able to
support the ever increasing demands of future Internet expansion. Fast,
stable data transmission is critically important to a wide array of
sectors of national interest, ranging from banking, the stock exchange,
and insurance to health services, transportation, and homeland security.
The information gleaned through the proposed research will aid in the
design and implementation of novel optical systems that will help meet the
growing need for bandwidth.
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| NSF Org: |
DMS - Division of Mathematical Sciences |
| Award Number: |
0505681 |
| Award Instrument: |
Standard Grant |
| Program Manager: |
Kenneth J. Shaw
DMS Division of Mathematical Sciences
MPS Directorate for Mathematical & Physical Sciences
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| NSF Program(s): |
APPLIED MATHEMATICS |
| Field Application(s): |
Other nsf.applications NEC |
| Program Reference Code(s): |
UNASSIGNED, 0000 |
| Program Element Code(s): |
1266 |
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