Small-amplitude grain boundaries of arbitrary angle in the Swift-Hohenberg equation

We study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of different orientations. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angles between the roll solutions. This extends prior work in [6] where the analysis was restricted to large angles, that is, weak bending near the grain boundary. The main new difficulty stems from possible interactions of the primary modes with other resonant modes. We generalize the normal form analysis in [6] and develop a singular perturbation approach to treat resonances.