0.0262$. The top row depicts intensity and phase distribution at different propagation distance while the bottom plots illustrate the full 3D evolution of the “soliton”. The rotation (counter-rotating w.r.t. phase) accompanied by beam deformation is clearly visible. For an amplitude ratio $B$ less than the critical value ($B<B_{\rm cr}$) the azimuthon no longer rotates but swings back and forth and hence its average rotation frequency is zero. The propagation of this wobbling dipole with an amplitude ratio $B$ smaller than the critical value is displayed in Fig. \[square\_t\_evolution\]. The right panels show the maximum angle which the dipole attains during propagation. The 3D surface plot in the bottom row clearly illustrates the swinging movement of the “soliton” upon propagation.

![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"}](sdt0 "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 