in the medium; $n_2$ is the Kerr nonlinear coefficient, $n(\xi,\eta)$ the linear refractive index distribution, $n_b$ the background index, and $\lim_{\xi,\eta \rightarrow \infty}n(\xi,\eta)=n_b$. We consider a weakly-guiding waveguide, so both the linear and nonlinear induced index change $|n-n_b|$ and $n_2|\mathcal{E}|^2$ are small compared to the mean index $n_b$, and at the same time $n_2|\mathcal{E}|^2 \ll |n-n_b|$, to guarantee a weak nonlinearity.

From the mathematical point of view, Eq. (\[physicaleq\]) is a (2+1)-dimensional nonlinear Schr[ö]{}dinger (NLS) equation with linear potential representing the waveguide profile. For technical convenience, we rescale Eq. (\[physicaleq\]) to dimensionless quantities with $x=\xi/r_0$, $y=\eta/r_0$, $z=\zeta/(2 k_0 r_0^2)$, and $\sigma=\mathrm{sgn}(n_2)$, where $r_0$ represents the transverse spatial extent of the waveguide. Then, the two-dimensional (2D) NLS equation for the scaled wave function $\psi = k_0 r_0 \sqrt{2|n_2|/n_b} \mathcal{E}$ reads $$i\frac{\partial}{\partial z} \psi + \left( \frac{\partial ^2}{\partial x^2}
 + \frac{\partial ^2}{\partial y^2} \right) \psi +
\sigma |\psi|^2 \psi + V \psi=0,
\label{partial}$$ with an “attractive” bounded potential $V=2k_0^2 r_0^2 (n-n_b)/n_b$, given by the spatial profile of the refractive index of the waveguide. Here we will consider propagation in step index waveguides with parameter $V(x,y)$ given as follows $$V(x,y)=
\begin{cases}
V_0 & \mathrm{~where~}
\begin{cases}
\sqrt{x^2+y^2} \leq 1 & \mathrm{~for~circular~waveguide}
\\
|x| \leq 