From the defocusing nonlinear Schroedinger to the complex Ginzburg-Landau equation

Abstract

Perturbation approaches developed so far for the dark soliton solutions of the (fully integrable) defocusing nonlinear Schroedinger equation cannot describe the dynamics resulting from dissipative perturbations of the Ginzburg-Landau type. Here spatially slowly decaying changes of the background wavenumber occur which requires the use of matching technics. It is shown how the perturbation selects a 1 or 2-parameter subfamily from the 3-parameter family of dark solitons of the nonlinear Schroedinger equation. The dynamics of the perturbed system can then be described analytically as motion within this selected subfamily yielding interesting scenarios. Interaction with shocks occurring in the complex Ginzburg-Landau equation can be included in a straight forward way.