Abstract
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.
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Hiraoka, Y., Ogawa, T. Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation. Japan J. Indust. Appl. Math. 22, 57 (2005). https://doi.org/10.1007/BF03167476
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DOI: https://doi.org/10.1007/BF03167476