Recovering radial amplitude-equation results via Galerkin finite-Fourier approximation

Determine whether the radial amplitude equations for fully localized dihedral patterns in the planar Swift–Hohenberg equation and in two-component reaction–diffusion systems—derived here using nonautonomous multiple-scale analysis with Bessel differential operators and convolutional Bessel identities—can be recovered from a Galerkin finite-Fourier approximation in the angular variable, despite the loss of those nonlinear Bessel identities in the truncated setting.

Background

The paper develops a formal multiple-scales framework in polar coordinates that yields radial amplitude equations for fully localized stripes, hexagons, rhomboids, and twelve-fold quasipatterns. A key ingredient is the use of Bessel differential operators and convolutional identities for products of Bessel functions, which enable simplification of nonlinear terms after angular Fourier projection.

The authors note that previous work considered Galerkin finite-Fourier truncations in the angular variable to paper localized dihedral patterns by reducing to large systems of radial ODEs. However, the present analysis crucially relies on convolutional Bessel identities that are not obviously preserved under finite truncation, raising the explicit question of whether the same amplitude equations (and corresponding localized solutions) can be obtained within a Galerkin finite-Fourier framework.