of higher order azimuthons constructed by using pairs of degenerated higher order waveguide modes. In particular, in a circular waveguide degenerate higher order modes have the form of quadrupoles, hexapoles, octapoles, decapoles, etc. and other modes with even number of lobes (optical necklaces). In a square waveguide, one can identify pairs of degenerate modes in a form of hexapoles, octapoles, decapoles, dodecapoles, etc. (optical matrices). Interestingly, the quadrupoles found in square waveguides are not degenerate and hence the propagation dynamics of their corresponding superposed states in weakly nonlinear regime is dominated by linear mode beating.

![The propagation of a hexapole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. Left panel shows the original hexapole azimuthon and the corresponding phase; right panel shows the iso-surface plot of the propagation.[]{data-label="cquadr_evolution"}](chexa0 "fig:"){width="45.00000%"} ![The propagation of a hexapole azimuthon with $A=0.4,~B=0.5$ in the circular waveguide. Left panel shows the original hexapole azimuthon and the corresponding phase; right panel shows the iso-surface plot of the propagation.[]{data-label="cquadr_evolution"}](chexa_prop "fig:"){width="50.00000%"}

Fig. \[cquadr\_evolution\] displays an example of hexapole azimuthon in a circular waveguide. The left panel represents the initial intensity and phase distribution of the azimuthon. The 3D surface plot in the right panel depicts stable and smooth rotation of 