Snaking without subcriticality: grain boundaries as non-topological defects

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.

Automated summary

This study uses the Swift-Hohenberg equation to describe nonlinear equilibria containing grain boundaries in spatial patterns. These grain boundaries consist of closed curves with penta-hepta defects, resulting from local orientation mismatches between competing hexagonal structures. Multiple isolas are obtained across a wide range of parameters, even without a unique Maxwell point.