Deformation of Striped Patterns by Inhomogeneities

TL;DR

This paper investigates how localized inhomogeneities affect striped pattern formations in a two-dimensional Ginzburg-Landau system, revealing a phase correction mechanism and establishing a mathematical framework for pattern deformation analysis.

Contribution

It introduces Kondratiev spaces to analyze inhomogeneity effects, proving Fredholm properties and constructing deformed patterns with a phase-selection mechanism.

Findings

Logarithmic phase correction identified

Kondratiev spaces effectively handle inhomogeneity effects

Deformed patterns constructed using the Implicit Function Theorem

Abstract

We study the effects of adding a local perturbation in a pattern forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity.