with the single charged vortex soliton [@lopez-aguayo_ol_2006]. A single charged vortex consists of two equal-amplitude dipole-shaped structures representing real and imaginary part of $U$. If these two components differ in amplitude, the resulting structure forms a “rotating dipole” azimuthon. If one of the components is zero we deal with the dipole soliton, which consists of two out-of-phase humps and does not rotate for symmetry reasons. In a first attempt, let us assume we know the radial shape of the linear vortex mode $F(r)$ which is normalized according to $\pi \int r|F(r)|^2 \mathrm{d}r =1$. Then, using separation of variables, we consider the simplest so-called “rotating dipole” azimuthon with ansatz [@stefan_oe_2008] $$U(r,\phi-\omega z)=AF(r)\left[\cos(\phi-\omega z)+i B \sin(\phi-\omega z) \right],
\label{complexamplitude}$$ where $A$ is an amplitude factor, and $1-B$ the azimuthal modulation depth of the resulting ring-like structure. Because we are operating in the weakly nonlinear regime, using linear waveguide modes as initial conditions for nonlinear (soliton) solutions is a quite good approximation.

After plugging Eq. (\[ansatz\]) into Eq. (\[partial\]), multiplying by $U^*$ and $\partial U^* / \partial \phi$ respectively, and integrating over the transverse coordinates we end up with [@stefan_oe_2008]

$$-\kappa P + \omega L_z + I + N = 0,$$

$$-\kappa L_z 