Snaking without subcriticality: grain boundaries as non-topological defects

TL;DR

This paper models grain boundaries as localized defects within pattern-forming systems using the Swift--Hohenberg equation, revealing stable structures without subcritical bifurcations and emphasizing the significance of pinning effects.

Contribution

It introduces a novel approach to treat non-topological defects as localized structures embedded in patterns, bypassing amplitude-phase decomposition and highlighting the role of pinning.

Findings

Stable grain boundary states with multiple defects are found.

Multiple isolated solution branches (isolas) exist across parameters.

Pinning effects are crucial for understanding defect dynamics.

Abstract

Non-topological defects such as grain boundaries abound in pattern forming systems, arising from local variations of pattern properties such as amplitude, wavelength, orientation, etc. We introduce the idea of treating such non-topological defects as spatially localised structures that are embedded in a background pattern, instead of treating them in an amplitude-phase decomposition. Using the two-dimensional quadratic-cubic Swift--Hohenberg equation as an example we obtain fully nonlinear equilibria that contain grain boundaries which are closed curves containing multiple penta-hepta defects separating regions of hexagons with different orientations. These states arise from local orientation mismatch between two stable hexagon patterns, one of which forms the localised grain and the other its background, and do not require a subcritical bifurcation connecting them. Multiple robust isolas that span 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 amplitude-phase description.