Grain boundaries in the Swift–Hohenberg equation

Abstract

We study the existence of grain boundaries in the Swift-Hohenberg equation. The analysis relies on a spatial dynamics formulation of the existence problem and a centre-manifold reduction. In this setting, the grain boundaries are found as heteroclinic orbits of a reduced system of ODEs in normal form. We show persistence of the leading-order approximation using transversality induced by wavenumber selection.

Continuation and Bifurcation of Grain Boundaries in the Swift--Hohenberg Equation

“…In a simplistic picture, one can envision effective gradient dynamics on the circle of grain boundaries parameterized by the relative phase, with at least two critical points. The proofs in [16] show that even and odd (in x) grain boundaries persist, with a phasemismatch of 0, π, respectively, at x = 0. We computed odd grain boundaries and showed that they possess properties similar to even grain boundaries, that is, they select zigzag marginally stable stripes; see Figure 15.…”

“…These methods have proven useful not only to establish local existence, but also to classify and study stability, bifurcations, and interactions of coherent structures. This spatial-dynamics perspective has also been used to study existence of grain boundaries close to onset of a pattern-forming instability [14,16,40]. Second, far from onset of pattern formation, qualitative changes in the nature of grain boundaries have been observed and quantified, both theoretically and numerically in [10,11,31] using phase approximations.…”

“…Mathematically, the question of existence was answered quite comprehensively in [14,16,40]. There, existence of symmetric grain boundaries, k − x = −k + x , k − y = k + y was shown for µ sufficiently small and arbitrary angle ∠(k − , k + ).…”

“…The results in [14,16,40] examine this dynamical system using center-manifold reduction and normal form theory. The ill-posed dynamical systems (1.4) is reduced to an ordinary differential equation on a locally invariant manifold.…”

The Swift–Hohenberg Equation under Directional-Quenching: Finding Heteroclinic Connections Using Spatial and Spectral Decompositions

“…Recently, different research avenues have been exploited: several studies have been done using rigorous numerical analysis [MS13], harmonic analysis techniques [JS15, Jar15, BLBL12], variational techniques [Rab94], or more functional-analytic based techniques [MS18]. Still, many classes of problems remain unsolved, as that of asymmetrical grain boundaries, a case that does not seem to be directly amenable to the spatial dynamics techniques as presented in [HS12,SW14]; in this scenario the far/near decompositions we presented might be relevant for analytical results. For numerical results which exploit far/near (spatial) decomposition, see [LS17]).…”