nonlinear means that the nonlinear induced index change, which is proportional to the intensity of the optical beam, is small compared to the index profile (or trapping potential) of the waveguide. Following [@stefan_pre_2004], we can expect that weakly nonlinear azimuthons are stable in multi-mode waveguides.

The paper is organized as follows: In Sec. \[modelling\], we briefly introduce the general model equation for beam propagation in weakly nonlinear waveguides, and then we discuss in detail the properties of dipole azimuthons in circular and square waveguides in Sec. \[dipoles\], respectively. In Sec. \[higherorder\], higher order azimuthons are investigated, and in Sec. \[conclusion\] we conclude.

Mathematical modeling {#modelling}
=====================

We consider the evolution of a continuous wave (CW) optical beam with amplitude $\mathcal{E} (\xi, \eta, \zeta)$, where $(\xi,\eta)$ and $\zeta$ denote the transverse and longitudinal coordinates, respectively. Then the propagation of this beam in a weakly-guiding waveguide with Kerr nonlinearity in the scalar, slowly varying envelope approximation is described by the following equation [@boyd_book_2008]: $$i \frac{\partial}{\partial \zeta} \mathcal{E} +
\frac{1}{2k_0} \left( \frac{\partial ^2}{\partial \xi^2} + \frac{\partial ^2}{\partial \eta^2} \right) \mathcal{E} +
k_0 \frac{n_2}{n_b} |\mathcal{E}|^2 \mathcal{E} + k_0 \frac{n(\xi,\eta)-n_b}{n_b} \mathcal{E}=0,
\label{physicaleq}$$ where $k_0=2 \pi n_b / \lambda_0$ refers to the carrier central wave number 