Reducing dimensionality to model 2D rotating and standing waves in a delayed nonlinear optical system with thin annulus aperture

The purpose of the paper is to theoretically investigate 2D wave patterns in a nonlinear optical system with diffractive feedback. We consider a delayed functional differential diffusion equation on a thin annulus with Neumann boundary conditions. To study pattern formation phenomena, we construct the Faria normal form of a Hopf bifurcation, which is necessarily degenerate because the equation exhibits O(2) symmetry. The coefficients of the normal form determine the excitation of rotating or standing waves with the prescribed spatio-temporal characteristics, but these coefficients can be expressed only implicitly. However, for sufficiently thin annuli, the equation corresponds to a limit 1D model on a circle, which was carefully analysed in our previous papers and whose normal form was explicitly calculated. 2D wave stability predictions that we make based on the normal form of the limit 1D model are in good agreement with direct numerical simulations of the 2D model. To accelerate computation, we elaborated an efficient fast finite Hankel transform algorithm for thin annular domains.