Pattern Formation on a Finite Disk, Variational and Non Variational Case #

Nicolas Verschueren van Rees, Edgar Knobloch, Hannes Uecker

14:10 Monday in 2Q49.

Part of the Dynamics of reaction-transport systems session.

Abstract #

The dynamics of the real cubic-quintic Swift-Hohenberg equation over a finite disk with no-flux boundary conditions are studied. The stability properties of the trivial state are determined via linear stability analysis. The unstable modes are followed via numerical continuation, revealing a variety of spatially extended and localized states. We compute families of solutions localized in the interior (multiarm or azimuthal), and at the periphery (localized wall modes). We identify the mechanisms by which localized solutions connect to a domain-filling state.

A nonvariational generalization, the complex cubic-quintic Swift-Hohenberg equation, is also considered. In this model, the trivial state becomes unstable via a Hopf bifurcation, generating standing and traveling waves. The associated mode can be spatially extended or take the form of an oscillatory wall mode. Standing oscillations of the latter type may be azimuthally periodic or azimuthally localized and resemble the corresponding states in the one-dimensional case with periodic boundary conditions. The relative stability of extended standing and traveling states is consistent with the predictions of a symmetry-breaking Hopf bifurcation with O(2) symmetry. These findings of this study are expected to be relevant in bistable pattern-forming systems on a disk, such as nonlinear optical systems, and low Prandtl number convection.