in circular waveguide rotating with $\omega \approx 0.037$ is illustrated in Fig. \[circularevolution\]. The top iso-surface plot displays 3D rigid and continuous rotation of the azimuthon during propagation. The row below depicts the transverse intensity (left) and phase (right) distribution of the dipole azimuthon during propagation after rotating by $\pi/2$.

![Azimuthon rotation frequency $\omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel). Blue solid lines show analytical predictions from Eq. (\[analyticalangular\]). Red dots denote results obtained from numerical simulations of Eq. (\[partial\]). The values of $A$ and $B$ are shown next to the lines.[]{data-label="circularrelations"}](relation_ca "fig:"){width="40.00000%"} ![Azimuthon rotation frequency $\omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel). Blue solid lines show analytical predictions from Eq. (\[analyticalangular\]). Red dots denote results obtained from numerical simulations of Eq. (\[partial\]). The values of $A$ and $B$ are shown next to the lines.[]{data-label="circularrelations"}](relation_cb "fig:"){width="40.00000%"}

![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cd_prop 