Graduate School in Nonlinear Science

Sponsored by The Danish ResearchAgency

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


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.