Graduate School in Nonlinear Science

Sponsored by The Danish Research Agency

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 29424

MIDIT-seminar 492

Tuesday, July 3, 2001, 14.00 h
at IMM, Bldg. 305, Room 018, DTU

Abstract: The possibility has been discussed of substantial highly localized dissipation from nonlinear waves through a combination of self focussing and nonlinear dissipation, as modelled by a Dissipative Nonlinear Equation. This has been based on the existence, in certain parameter regimes, of stationary singular solutions of the nondissipative NLSE having dissipation at the focussing singularity, which could provide approximations of behaviour in the small dissipation limit.
Unfortunately, these singular solutions do not exist in the main physical cases such as (1+3) dimensions with up to cubic nonlinearity, and (1+2) dimensions with any nonlinearity power, and in these cases, numerical simulations show that dissipation causes focii to collapse so rapidly that there total dissipation goes to zero as the dissipation coefficient does.
Numerical studies show that this also tends to happen for some cases where the singular dissipative solutions do exist. However the studies also show that focii can repeatedly reform after collapse, leading to a rapid train of dissipative focii and the possibility of substantial total dissipation.
Studying the space and time structure of this multi-focussing leads to some ideas about when, whether and how the total dissipation can be significant as the dissipation coefficient approaches zero.