Nicolas Verschuren van Rees (University of Exeter) – Localised, Extended and Dynamic Patterns on a Finite Disk

ABSTRACT

We study both a variational and a non-variational (complex) version of the cubic-quintic Swift–Hohenberg model (SH35) in a finite disk with Neumann boundary conditions. These prototype models are relevant in contexts such as fluid dynamics, combustion experiments, and nonlinear optics.

Using a combination of numerical methods (direct numerical simulations and continuation techniques via pde2path) and analytical approaches (linear and weakly nonlinear analysis), we identify and study branches of extended and localised patterns. In the variational case, we find three qualitatively distinct types of branches termed: wall modes, axisymmetric modes, and extended states. For each branch type, secondary bifurcations give rise to localised states organised in an extension of the so-called snaking scenario.

In the non-variational case, many of the observed branches correspond to dynamical counterparts of the variational branches. The system supports travelling, modulated travelling, standing, and localised standing waves. Due to numerical constraints on the disk radius, we also analyse a one-dimensional version of the non-variational problem with periodic boundary conditions. This reduced problem provides a good description of the organisation of wall-mode solutions, enabling analytical tractability via weakly nonlinear methods.