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Rigorous numerics for localized patterns to the quintic Swift-Hohenberg equation. (English) Zbl 1067.65146
Summary: 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.

65P30 Numerical bifurcation problems
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
37M20 Computational methods for bifurcation problems in dynamical systems
C-XSC; C-XSC 2.0
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