Modulated and Localized States in a Finite Domain

A subcritical pattern-forming (Turing) instability of a uniform state, in an infinite domain, produces two branches of spatially localized states that bifurcate from the pattern-forming instability along with a uniform spatially periodic pattern. In this paper we demonstrate that branches of localized states persist as strongly amplitude-modulated patterns in large, but finite, domains with periodic boundary conditions. Our analysis is carried out for a model Swift–Hohenberg equation with a cubic-quintic nonlinearity. If the domain size exceeds a critical value, modulated states appear in secondary bifurcations from the primary branch of spatially periodic solutions. Multiple-scales analysis indicates that these secondary bifurcations occur close to the primary instability and close to the saddle-node bifurcation on the spatially periodic solution branch. As the domain size increases, extra “turns” on the snaking curve arise through a repeating sequence of saddle-node bifurcations and mode interactions between periodic patterns of nearby wavenumbers. With periodic boundary conditions, the crosslinks in the “snakes and ladders” structure identified by previous authors in the infinite domain case also persist in finite domains. However, the imposition of either Dirichlet or Neumann boundary conditions preserves one of the two branches of localized states that exist in the infinite domain case and causes the other to fragment.