Graduate School in Nonlinear Science
MIDIT OFD CATS
Modelling, Nonlinear Dynamics Optics and Fluid Dynamics Chaos and Turbulence Studies
and Irreversible Thermodynamics Risø National Laboratory Niels Bohr Institute and
Technical University of Denmark Building 128 Department of Chemistry
Building 321 P.O. Box 49 University of Copenhagen
DK-2800 Lyngby DK-4000 Roskilde DK-2100 Copenhagen Ø
Denmark Denmark Denmark
PREVENTING COLLAPSE AND BEAM DISPERAL IN THE NONLINEAR
SCHRÖDINGER EQUATION, USING ATTRACTIVE POTENTIALS
by Brenton LeMesurier
Department of Mathematics,
College of Charleston
Charleston SC, USA
MIDIT-seminar 506
Thursday, May 30, 2002, 15.00 h
at IMM, Bldg. 305, Room 130, DTU
Abstract:
The 2D Focusing Cubic Nonlinear Schrödinger Equation (NLS) models laser
beam propagation and other nonlinear wave phenomena. For initial data
of Gaussian cross section, one will have point singularity formation
through self-focusing collapse for sufficiently intense initial beams,
and for less intense beams one typically sees dispersal.
Adding an attractive finite range potential such as one of Gaussian form
can give a model (NLSGP) of a doped central core in the propagation
medium, and also an improvment on the quadratic "trap" potential in the
Gross-Pitaevski model of Bose-Einstein condensates.
Numerical simulations of NLSGP will be presented which show,
surprisingly, that a sufficiently narrow and deep potential can prevent
collapse, instead trapping the beam into narrow oscillations lying
mostly within the potential well. Weaker initial beams that would
disperse without the potential are also seen, less suprisingly, to be
trapped into such oscillations.
Analytical work in progress will be discussed on possible mechanisms of
radiation giving relaxation towards orbitally stable steady states of
this NLS equation with potential.