Snaking without subcriticality: grain boundaries as non-topological defects

• Published in: IMA journal of applied mathematics Vol. 86; no. 5; pp. 1164 - 1180
• Main Authors: Subramanian, Priya, Archer, Andrew J, Knobloch, Edgar, Rucklidge, Alastair M
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01.10.2021
• Subjects: non-topological defect; penta-hepta defect; grain boundaries; spatial localization; Swift–Hohenberg equation
• ISSN: 0272-4960; 1464-3634
• DOI: 10.1093/imamat/hxab032

Abstract Non-topological defects in spatial patterns such as grain boundaries in crystalline materials arise from local variations of the pattern properties such as amplitude, wavelength and orientation. Such non-topological defects may be treated as spatially localized structures, i.e. as fronts connecting distinct periodic states. Using the two-dimensional quadratic-cubic Swift–Hohenberg equation, we obtain fully nonlinear equilibria containing grain boundaries that separate a patch of hexagons with one orientation (the grain) from an identical hexagonal state with a different orientation (the background). These grain boundaries take the form of closed curves with multiple penta-hepta defects that arise from local orientation mismatches between the two competing hexagonal structures. Multiple isolas occurring robustly over a wide range of parameters are obtained even in the absence of a unique Maxwell point, underlining the importance of retaining pinning when analysing patterns with defects, an effect omitted from the commonly used amplitude-phase description. Similar results are obtained for quasiperiodic structures in a two-scale phase-field model.