To snake or not to snake in the planar Swift-Hohenberg Equation

Daniele Avitabile*, David J.B. Lloyd, John Burke, Edgar Knobloch, Björn Sandstede

*Corresponding author for this work

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We investigate the bifurcation structure of stationary localized patterns of the two-dimensional Swift-Hohenberg equation on an infinitely long cylinder and on the plane. On cylinders, we find localized roll, square, and stripe patches that exhibit snaking and nonsnaking behavior on the same bifurcation branch. Some of these patterns snake between four saddle-node limits; in this case, recent analytical results predict the existence of a rich bifurcation structure to asymmetric solutions, and we trace out these branches and the PDE spectra along these branches. On the plane, we study the bifurcation structure of fully localized roll structures, which are often referred to as worms. In all the above cases, we use geometric ideas and spatial-dynamics techniques to explain the phenomena that we encounter.

Original languageEnglish
Pages (from-to)704-733
Number of pages30
JournalSIAM Journal on Applied Dynamical Systems
Volume9
Issue number3
DOIs
Publication statusPublished - 25 Oct 2010
Externally publishedYes

Keywords

  • Localized structures
  • Planar patterns
  • Snaking
  • Swift-Hohenberg

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