Inhomogeneities in 3 dimensional oscillatory media

TL;DR

This paper investigates how localized phase-changing heterogeneities affect three-dimensional oscillatory media modeled by the complex Ginzburg-Landau equation, revealing new spectral properties and solution behaviors.

Contribution

It demonstrates that considering Kondratiev spaces makes the linearization a Fredholm operator, enabling the construction of solutions near equilibrium and asymptotic analysis of wavenumbers.

Findings

Linearization is Fredholm in Kondratiev spaces.

Construction of solutions close to equilibrium.

Asymptotic behavior of wavenumbers in the far field.

Abstract

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.