Localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk

TL;DR

This paper investigates various localized and extended patterns in the cubic-quintic Swift-Hohenberg equation on a disk, revealing complex bifurcation structures and boundary interactions through numerical analysis.

Contribution

It provides a detailed bifurcation analysis of localized and extended patterns in the Swift-Hohenberg equation on a disk, including new types of states like daisies and worms.

Findings

Localized spots and rings persist and snake until boundary interaction.

Secondary instabilities lead to multi-arm localized structures.

Discovery of daisy, worm, and stripe states with complex bifurcations.

Abstract

Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multi-arm localized structures that grow and interact with the boundary before broadening into domain filling states. High azimuthal wavenumber wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.