The phase structure of grain boundaries

This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical pattern-forming system, the Swift-Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross-Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed.This article is part of the theme issue 'Stability of nonlinear waves and patterns and related topics'.