The following is a list of my publications.
Papers:
- Existence, Stability, and Scattering of Bright Vortices in the Cubic-Quintic Nonlinear Schrödinger
Equation.
R.M.
Caplan,
R.
Carretero-González, P.G.
Kevrekidis, and B.A.
Malomed.
SUBMITTED: Mathematics
and Computers in Simulation.
Abstract.
PDF.
- Azimuthal
Modulational
Instability of Vortices in the Nonlinear Schrödinger Equation.
R.M.
Caplan,
Q.E.
Hoq,
R.
Carretero-González, and
P.G.
Kevrekidis.
Optics
Communications. 282
(2009) 1399-1405.
Abstract.
PDF.
Masters
Thesis:
Azimuthal
Modulational
Instability of Vortex Solutions to the Two Dimensional Nonlinear
Schrödinger
Equation
Masters
Thesis in Computational
Science.
Thesis chair: Professor
Ricardo Carretero.
Abstract:
We
study the
azimuthal modulational instability (MI) of vortices with different
topological
charges, in the focusing two-dimensional nonlinear Schrödinger
(NLS)
equation. This setting has direct application in the realm of
Bose-Einstein condensates and light propagation in nonlinear
crystals.
The method of studying the stability relies on freezing the radial
direction in
the Lagrangian functional of the NLS in order to form a
quasi-one-dimensional
azimuthal equation of motion, and then applying a stability analysis in
Fourier
space of the azimuthal modes. We formulate predictions of growth rates
of
individual modes and find that vortices are unstable below a critical
azimuthal
wave number. Steady state vortex solutions are found by first
using a
variational approach to obtain an asymptotic analytical ansatz, and
then using
it as an initial condition to a nonlinear equation numerical
optimization
routine. The stability analysis predictions are corroborated by direct
numerical simulations of the NLS performed on a polar coordinate
finite-difference scheme. We briefly show how to extend the
method to
encompass nonlocal nonlinearities that tend to stabilize solutions.
