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.