Pinning and dipole asymptotics of locally deformed striped phases

TL;DR

This paper studies how spatial inhomogeneities influence striped patterns, revealing that they select specific pattern positions and cause algebraic dipole-like deformations characterized by effective moments and Green's function derivatives.

Contribution

It introduces a novel analytical framework combining mode filters, conjugacies, and Kondratiev spaces to analyze inhomogeneity effects on striped phases, including farfield decay and phase shifts.

Findings

Inhomogeneities select specific pattern translations.

Induction of algebraic dipole-type farfield deformations.

Farfield decay proportional to Green's function derivatives.

Abstract

We investigate the effect of spatial inhomogeneity on perfectly periodic, self-organized striped patterns in spatially extended systems. We demonstrate that inhomogeneities select a specific translate of the striped patterns and induce algebraically decaying, dipole-type farfield deformations. Phase shifts and leading order terms are determined by effective moments of the spatial inhomogeneity. Farfield decay is proportional to the derivatives of the Green's function of an effective Laplacian. Technically, we use mode filters and conjugacies to an effective Laplacian to establish Fredholm properties of the linearization in Kondratiev spaces. Spatial localization in a contraction argument is gained through the use of an explicit deformation ansatz and a subtle cancellation in Bloch wave space.