boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are $A=0.4,~B=0.2$.[]{data-label="square_t_evolution"}](sdt40 "fig:"){width="35.00000%"}\
![The propagation of a dipole azimuthon that wobbles in the square waveguide. The first row depicts the dipole at input and maximum displacement, respectively. The white line indicates the waveguide boundaries. The iso-surface plot below displays the twist and deformation during propagation. The chosen amplitude factor and ratio are $A=0.4,~B=0.2$.[]{data-label="square_t_evolution"}](sdt_prop "fig:"){width="55.00000%"}

Due to the azimuthon profile deformation, it is not possible to find an analytical expression of $\bar \omega$ as a function of $A$ and $B$ as in the case of circular waveguide (i.e. Eq. (\[analyticalangular\])). Therefore, we need to resort to numerical simulations. In Fig. \[squarerelations\] we show the numerically determined relation between $\bar \omega$ and the azimuthon parameters $A$ (left panel) and $B$ (right panel). The threshold-like behavior is evident in the right graph of this figure: the region $B<B_{\rm cr}$, in which the soliton wobbles, is depicted by a green line.

![Averaged rotation frequency $\bar \omega$ versus amplitude factor $A$ (left panel) and amplitude ratio $B$ (right panel) in a square waveguide. Red dots are numerical results obtained from Eq. (\[partial\]), blue curves ($B<0.4$) and blue 