Characterizing the effect of boundary conditions on striped phases

TL;DR

This paper investigates how boundary conditions affect stationary periodic patterns in one-dimensional systems, using theoretical analysis and computational methods to understand equilibrium structures and their dependence on boundary types.

Contribution

It introduces a framework based on displacement-strain curves to analyze boundary effects on equilibria, distinguishing between wavenumber- and phase-selecting conditions, with applications to various models.

Findings

Boundary conditions significantly influence the set of equilibria in large domains.

Displacement-strain curves effectively characterize boundary layer effects.

Continuation methods can compute these curves in complex systems like Swift-Hohenberg.

Abstract

We study the influence of boundary conditions on stationary, periodic patterns in one-dimensional systems. We show how a conceptual understanding of the structure of equilibria in large domains can be based on the characterization of boundary layers through displacement-strain curves. Most prominently, we distinguish wavenumber-selecting and phase-selecting boundary conditions and show how they impact the set of equilibria as the domain size tends to infinity. We illustrate the abstract concepts in the phase-diffusion and the Ginzburg-Landau approximation. We also show how to compute displacement-strain curves in more general systems such as the Swift-Hohenberg equation using continuation methods.