"fig:"){width="55.00000%"}\
![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cdipole0 "fig:"){width="35.00000%"} ![The propagation of a dipole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. The iso-surface plot on the top clearly displays the spiraling of the azimuthon during propagation. Figures in the row below depict the azimuthon’s transverse intensity distribution at input, and after rotating $\pi/2$, respectively. The white line indicates the waveguide boundaries.[]{data-label="circularevolution"}](cdipole85 "fig:"){width="35.00000%"}

Azimuthon-like dipoles in square waveguides
-------------------------------------------

Because of the lack of circular symmetry, azimuthons in the strict sense of Eq. (\[complexamplitude\]) (preservation of shape during rotation with constant frequency) cannot exist in a square waveguide. However, as we will show below, the azimuthon-like behavior is still possible even though the beam propagation is accompanied by variation of the beam transverse intensity distribution. To set the initial condition, we use two linear degenerated orthogonal dipole modes $D_1$, $D_2$ (as in the case of the circular waveguide), which are normalized according to $\iint |D_{1,2}|^2~\mathrm{d}x~\mathrm{d}y=1$, and 