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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Scheel, Arnd Lloyd, David J. B.
Copyright Year	2017
Abstract	We study grain boundaries between striped phases in the prototypical Swift--Hohenberg equation. We propose an analytical and numerical far-field-core decomposition that allows us to study existence and bifurcations of grain boundaries analytically and numerically using continuation techniques. This decomposition overcomes problems with computing grain boundaries in a large doubly periodic box with phase conditions. Using the spatially conserved quantities of the time-independent Swift--Hohenberg equation, we show that symmetric grain boundaries must select the marginally zig-zag stable stripes. We find that as the angle between the stripes is decreased, the symmetric grain boundary undergoes a parity-breaking pitchfork bifurcation where dislocations at the grain boundary split into disclination pairs. A plethora of asymmetric grain boundaries (with different angles of the far-field stripes on either side of the boundary) is found and investigated. The energy of the grain boundaries is then mapped out. We find that when the angle between the stripes is greater than a critical angle, the symmetric grain boundary is energetically preferred, while when the angle is less than the critical angle, grain boundaries where stripes on one side are parallel to the interface are energetically preferred. Finally, we propose a classification of grain boundaries that allows us to predict various nonstandard asymmetric grain boundaries.
Sponsorship	Faculty Research Support Fund. WestfÃ¤lische Wilhelms-UniversitÃ¤t MÃ¼nster. DAAD Faculty Research Visit Grant. Institute for Mathematics and Its Applications. London Mathematical Society. National Science Foundation
Starting Page	252
Ending Page	293
Page Count	42
File Format	PDF
DOI	10.1137/16M1073212
e-ISSN	15360040
Issue Number	1
Volume Number	16
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2017-01-31
Access Restriction	Subscribed
Subject Keyword	Pattern formation spatial dynamics grain boundaries Bifurcations and instability continuation
Content Type	Text
Resource Type	Article

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