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

Lloyd, David and Scheel, A (2017) Continuation and Bifurcation of Grain Boundaries in the Swift-Hohenberg Equation SIAM Journal on Applied Dynamical Systems, 16 (1). pp. 252-293.

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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 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, the 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 non-standard asymmetric grain boundaries.

Item Type: Article
Subjects : Mathematics
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Lloyd, DavidD.J.Lloyd@surrey.ac.ukUNSPECIFIED
Scheel, AUNSPECIFIEDUNSPECIFIED
Date : 13 February 2017
Identification Number : 10.1137/16M1073212
Copyright Disclaimer : Copyright 2017 Society for Industrial and Applied Mathematics
Related URLs :
Depositing User : Symplectic Elements
Date Deposited : 24 Oct 2016 10:26
Last Modified : 19 Jul 2017 10:46
URI: http://epubs.surrey.ac.uk/id/eprint/812563

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