the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr0 "fig:"){width="35.00000%"} ![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr30 "fig:"){width="35.00000%"}\
![The propagation of the azimuthon-like dipole, that rotates in the square waveguide. Top row: intensity (left) and phase (right) distribution corresponding to the input and soliton rotation by $\pi/4$, respectively. The white line indicates the waveguide boundaries. The iso-surface plot at the bottom displays the rotation and deformation during propagation. The initial amplitude and modulation parameters are $A=0.4,~B=0.5$.[]{data-label="square_r_evolution"}](sdr_prop "fig:"){width="55.00000%"}

Let us focus on the case where the initial dipole-like field structure is oriented along the diagonal of the waveguide cross-section, and from now on the notation $D_1$ and $D_2$ stands for this orientation. Figure \[square\_r\_evolution\] shows an exemplary propagation of the resulting azimuthon-like solution with $B>B_{\rm cr}$, and an average rotation frequency of $\bar \omega \approx 