Research statement
In general, my research lies at the interface of mathematics and
real-world physical phenomena.
It consists of modeling physical systems
in terms of ordinary and partial differential equations
and employing functional analysis,
asymptotic and perturbation analysis, and numerical computations
to analyze the models in detail.
The over-arching goal of my research is to connect between
the mathematical and physical aspects arising from these
problems and make reliable and useful predictions about physical
systems.
Nonlinear waves
Thus far, most of my research has focused on nonlinear waves.
Mathematically, this has revolved around nonlinear Schrodinger (NLS) equations.
Within the context of nonlinear optics, my research results have
application to the propagation of laser beams in optical fibers, air
and water, the control of ultrashort pulses, and the design of
ultrafast mode-locked lasers.
I have recently become interested in the dynamics of ultra-cold
atomic gases and in particular in Bose-Einstein condensates (BECs).
When matter is cooled down very near the absolute zero temperature,
a large fraction of the atoms collapse into the lowest allowable
quantum state, giving rise to quantum effects on a macroscopic
scale.
My research aims to help answer fundamental physical questions, such as how
these systems behave.
In-spite of their physical differences, the dynamics of intense optical
beams and atomic BECs can be described using suitable NLS
equations, also known as the Gross-Pitaevskii equation in the BEC context.
These equations gives rise to special solutions that are
localized nonlinear modes, called solitary waves or solitons.
In some regimes these waves can propagate through the medium almost
unscathed, while in other regimes they can undergo catastrophic
self-focusing (singularity formation).
This leads to many intriguing mathematical questions about the properties of
solitons. Understanding this dynamics will offer insight into
experimental results and help in the design of and control of these systems.
Solar science
Renewable energy resources are becoming increasingly important for the
sustainability and welfare of mankind. One of the most abundant
resources is our Sun. However, only a small fraction of electricity
comes directly form the Sun mainly because the difficulty to
convert solar energy to electricity. This can
be achieved using Photovoltaic (PV) cells, a technology that has undergone
vast improvements yet remains expensive. A potentially more cost-effective approach is
to use a luminescent solar concentrator (LSC) to collect light, guide
it and concentrate it onto a small PV cell. Along with my colleagues we are
attempting to develop a new kind of LSC based on semiconductor nanorods
that will be much more efficient and long-lasting than existing dye-based LSCs.
Mathematically, this project entails analytical and computational
modeling of the propagation, multiple absorption and scattering of
light in LSCs.
The results of these models will help inform the synthesis
efforts of the optimal design parameters.
This research is supported by a recent NSF SOLAR Award.
See also this
press release.
You are welcomed to browse my publications
and contact me if you are interested in learning
more about my research.
You may also download my Ph.D. thesis, which was submitted to the School of
Mathematical Sciences at Tel Aviv University in 2002.