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

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.