$J_1$ and modified Bessel function of second kind $K_1$: $$F(r) = C \times
\begin{cases}
J_1 ( \sqrt{V_0-\kappa} r) & \mathrm{for}~ 0\leq r\leq1
 \\
\dfrac{J_1(\sqrt{V_0-\kappa})}{K_1(\sqrt{\kappa})} \cdot K_1 (\sqrt{\kappa}r) & \mathrm{for}~ r>1
\end{cases},$$ where $C$ is a normalization factor such that $\pi\int|F(r)|^2rdr=1$. Thus, the analytically predicted rotation frequency is $$\omega=\frac{\pi \sigma}{2} \int r |F(r)|^4 \mathrm{d}r \cdot A^2 B.
\label{analyticalangular}$$ From the above equation, one can see that sense of rotation is opposite for focusing and defocusing nonlinearities. Moreover, as $\omega = 2 k_0 r_0^2 \cdot 2 \pi/ \ell$, one can find from Eq. (\[analyticalangular\]) an expression for the physical distance $\ell$ over which the azimuthon in a circular waveguide performs one full rotation: $$\ell= \frac{\displaystyle 8 k_0 r_0^2}{\displaystyle A^2 B \int r |F(r)|^4 \mathrm{d}r}.
\label{physicaldistance}$$ For example, assuming propagation in a silica fiber with the parameters discussed earlier (see Sec. \[modelling\]) and taking $A=0.4$, $B=0.5$, we obtain $\ell \approx 2.4 ~\mathrm{m}$.

Figure \[circularrelations\] shows the dependence of the azimuthon rotation frequency as a function of its amplitude $A$ (left panel) and the modulation parameter $B$ (right panel). Blue solid lines represent the analytical predictions and, in excellent agreement, red dots are obtained from numerical simulations. An exemplary propagation of the dipole azimuthon 