the radius of a standard circular multi-mode fiber [@agrawal_book_2009]. In addition, in order to guarantee intensities smaller than the LIDT the condition $|\psi|^2<1/3$ should be fulfilled.

Note that in Eq. (\[partial\]) $\sigma$ is the sign of nonlinearity: $\sigma =1$ ($\sigma=-1$) represents focusing (defocusing) nonlinearity. The main difference between weakly nonlinear waveguides with focusing and defocusing nonlinearity is that defocusing nonlinearity supports higher amplitudes of the wave function, and in general leads to more stable and robust configurations. In this paper, however, we consider the experimentally relevant case of a focusing nonlinearity.

Rotating localized dipoles {#dipoles}
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Dipole azimuthons in circular waveguides
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Azimuthons are a straightforward generalization of the usual ansatz for stationary solutions [@desyatnikov_prl_2005]. They represent spatially rotating structures and hence involve an additional parameter, the rotation frequency $\omega$ (see also [@skryabin_pre_2002]), so we seek approximate solutions of the form $$\psi (r,\phi,z) = U (r, \phi - \omega z) \exp (i \kappa z),
\label{ansatz}$$ where $r=\sqrt{x^2+y^2}$ and $\phi$ the azimuthal angle in the transverse plane $(x,y)$, $U$ is the stationary profile, $\omega$ the rotation frequency, and $\kappa$ the propagation constant. For $\omega=0$, azimuthons become ordinary (non-rotating) solitons. The simplest family of azimuthons is the one connecting the dipole soliton 