(a) Spatial distribution of particle density |ψ₀|² in the x–y plane. Annular solitons along the radial direction can be clearly observed with an approximate axial symmetry. From the surface plot at the top, we find that the density distribution acquires a slight axial modulation, reflecting the presence of background lattice potential with eightfold rotational symmetry. (b) Density and phase profiles along the x-direction. Notice that the phase (blue dashed-dotted) changes abruptly at the positions of zeros or minima of the particle density (black solid). In this plot, the rotating frequency Ω = π, while other parameters are the same as those in Fig. 5.