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 Mads Peter Sørensen
Informatics and Mathematical Modelling (IMM)
Technical University of Denmark
Building 321
2800 Kgs. Lyngby, Denmark

MIDIT-seminar 505

Thursday, January 31, 2002, 15.00 h
at IMM, Bldg. 305, Room 018, DTU

Abstract: The Maxwell equations coupled to a single Lorentz oscillator and with instantaneous Kerr nonlinearity are investigated. The existence of soliton-type solutions in the Schrödinger regime or light bullets containing few optical cycles together with dark solitons are illustrated numerically. Envelope collapse regimes of the Schrödinger equation are compared to the full system and an arrest mechanism is clearly identified when the spectral width of the initial pulse broadens beyond the applicability of the asymptotic behavior. We show that beyond certain threshold the carrier wave steepens into an infinite gradient similarly to the canonical Majda-Rosales weakly dispersive system. The weak dispersion in general cannot prevent the wave breaking with instantaneous or delayed nonlinearities.

Lagrangian and Hamiltonian formulations of the equations are determined. From Lie point symmetries admitted by the equations four conservation laws are obtained. The symmetries are used to obtain classical similarity solutions. In particular we have found a remarkable kink shaped optical travelling wave solution. Its stability has been assessed by numerical simulations. For typical physical parameter values the kink width is of order 10th of femtoseconds in strongly nonlinear optical polymers.