+ \omega P' + I' + N' = 0.$$

This system relates the propagation constant $\kappa$ and the rotation frequency $\omega$ of the azimuthons to integrals over their stationary amplitude profiles:

$$P = \iint r|U(r)|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$

$$L_z = -i\iint r\frac{\partial U(r)}{\partial \phi} U^*(r) ~\mathrm{d}r~\mathrm{d}\phi,$$

$$I = \iint rU^*(r) \Delta_{\perp} U(r) ~\mathrm{d}r~\mathrm{d}\phi,$$

$$N = \iint r\left[\sigma|U(r)|^2+V \right] |U(r)|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$

$$P' = \iint r\left| \frac{\partial U(r)}{\partial \phi} \right|^2 ~\mathrm{d}r~\mathrm{d}\phi,$$

$$I' = i\iint r\frac{\partial U^*(r)}{\partial \phi} \Delta_{\perp} U(r) ~\mathrm{d}r~\mathrm{d}\phi,$$

$$N' = i\iint r\left[\sigma|U(r)|^2+V \right] \frac{\partial U^*(r)}{\partial \phi} U(r) ~\mathrm{d}r~\mathrm{d}\phi.$$

The first two quantities $(P$ and $L_z)$ have straightforward physical meanings, namely power and angular momentum. The integrals $I$ and $I'$ are related to the diffraction mechanism of the system, whereas $N$ and $N'$ account for waveguide and nonlinearity. Thus we can formally solve for the rotation frequency with these quantities and obtain $$\omega=\frac{P(I'+N')-L_z(I+N)}{L_z^2-PP'}.
\label{angularfrequency}$$

After inserting Eq. (\[complexamplitude\]) into Eq. (\[angularfrequency\]), it turns out that only the nonlinear term contributes to the rotation frequency $\omega$ (see also Eq. (10) in Ref. [@fabian_pra_2010]). In order to give an estimate for the rotation frequency, we use the linear stationary modal profile $F(r)$ of a circular waveguide expressed in terms of Bessel functions of first kind 