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

Area of Science:

• Nonlinear Dynamics
• Pattern Formation
• Mathematical Physics

Background:

• The cubic-quintic Swift-Hohenberg equation models pattern formation in various physical systems.
• Understanding solutions on bounded domains, like disks, is crucial for realistic applications.
• Neumann boundary conditions are investigated for their influence on pattern stability.

Purpose of the Study:

• To investigate axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation on a disk.
• To analyze the behavior and stability of localized and domain-filling states.
• To identify and characterize novel pattern solutions, such as daisy and worm states.

Main Methods:

• Numerical continuation techniques were employed to trace solution branches.
• Bifurcation analysis was used to identify qualitative changes in solution behavior.
• The study focused on solutions within a disk geometry with Neumann boundary conditions.

Main Results:

• Axisymmetric localized states (spots, rings) were observed to persist and interact with the boundary.
• Secondary instabilities led to multiarm localized structures and domain-filling states.
• Novel high azimuthal wave number states, termed 'daisy states,' were discovered, with bifurcations yielding localized daisies, worms, and stripes.

Conclusions:

• The study reveals a rich variety of pattern solutions for the cubic-quintic Swift-Hohenberg equation on a disk.
• Boundary interactions and secondary instabilities play significant roles in pattern selection and evolution.
• The findings contribute to the understanding of nonlinear pattern formation in confined geometries.