All we know about hole solutions in the CGLE

The cubic Complex Ginzburg-Landau Equation (CGLE) has a one parameter family of traveling localized source solutions which become a subfamily of the dark soliton solutions in the nonlinear Schroedinger (NLS) limit. These so called "Nozaki-Bekki holes" are structurally unstable and a perturbation of the equation in general leads to a (positive or negative) monotonic acceleration or oscillation of the holes. This confirms that the cubic CGLE has an inner symmetry. As a consequence small perturbations change some of the qualitative dynamics of the cubic CGLE and lead to spatiotemporal intermittency in some parameter range. An analytic stability analysis of holes in the cubic CGLE and a semianalytical treatment of the acceleration in the perturbed equation is presented by using matching and perturbation methods. In the NLS limit fully analytical treatment is possible. Exploiting a phase conservation condition it is shown how the perturbation selects a 1 or 2 parameter subfamily from the 3 parameter family of dark solitons of the NLS equation. The dynamics of the perturbed system can then be described analytically as motion within the selected subfamily.