Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation

  • Yasuaki Hiraoka
  • Toshiyuki Ogawa


Localized patterns of the quintic Swift-Hohenberg equation are studied by bifurcation analysis and rigorous numerics. First of all, fundamental bifurcation structures around the trivial solution are investigated by a weak nonlinear analysis based on the center manifold theory. Then bifurcation structures with large amplitude are studied on Galerkin approximated dynamical systems, and a relationship between snaky branch structures of saddle-node bifurcations and localized patterns is discussed. Finally, a topological numerical verification technique proves the existence of several localized patterns as an original infinite dimensional problem, which are beyond the local analysis.

Key words

quintic Swift-Hohenberg equation localized patterns snaky bifurcation structure rigorous numerics Conley index 


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Copyright information

© JJIAM Publishing Committee 2005

Authors and Affiliations

  • Yasuaki Hiraoka
    • 1
  • Toshiyuki Ogawa
    • 1
  1. 1.Department of Mathematical Science, Graduate School of Engineering ScienceOsaka UniversityOsakaJapan

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