Continuation and bifurcation of grain boundaries in the swift-hohenberg equation

David J.B. Lloyd, Arnd Scheel

Research output: Contribution to journalArticlepeer-review

26 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)252-293
Number of pages42
JournalSIAM Journal on Applied Dynamical Systems
Issue number1
StatePublished - 2017

Bibliographical note

Funding Information:
The work of the first author was supported by the Institute for Mathematics and Its Applications and the Faculty Research Support Fund (University of Surrey). The work of the second author was partially supported by NSF grants DMS-0806614 and DMS-1311740, a DAAD Faculty Research Visit grant, and a WWU Fellowship. Both authors were supported by the London Mathematical Society through Research in Pairs grant 41502.

Publisher Copyright:
© 2017 Society for Industrial and Applied Mathematics.


  • Continuation
  • Grain boundaries
  • Spatial dynamics


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