  We consider localized perturbations to spatially homogeneous oscillations in
dimension 3 using the complex Ginzburg-Landau equation as a prototype. In
particular, we will focus on heterogeneities that locally change the phase of
the oscillations. In the usual translation invariant spaces and at $
\varepsilon=0$ the linearization about these spatially homogeneous solutions
result in an operator with zero eigenvalue embedded in the essential spectrum.
In contrast, we show that when considered as an operator between Kondratiev
spaces, the linearization is a Fredholm operator. These spaces consist of
functions with algebraical localization that increases with each derivative. We
use this result to construct solutions close to the equilibrium via the
Implicit Function Theorem and derive asymptotics for wavenumbers in the far
field.
