panel: The angular momenta of the rotating superposed dipoles (corresponding to Fig. \[square\_r\_evolution\], blue curve), and the twisting ones (corresponding to Fig. \[square\_t\_evolution\], red curve). The other curves depict the cases of $B$ slightly above (black) and bellow (green) $B_{\rm cr}$. All curves are obtained for $A=0.4$. []{data-label="angularmomentumcurve"}](hamil "fig:"){width="45.00000%"} ![ Left panel: Hamiltonian of $A(D_1+iBD_2)$ (red curve) as a function of $B$ and Hamiltonian of $D_3$ (blue line) (for the same power $P$). The intersection of the two curves defines $B_{\rm cr}$ (depicted by the black square). Right panel: The angular momenta of the rotating superposed dipoles (corresponding to Fig. \[square\_r\_evolution\], blue curve), and the twisting ones (corresponding to Fig. \[square\_t\_evolution\], red curve). The other curves depict the cases of $B$ slightly above (black) and bellow (green) $B_{\rm cr}$. All curves are obtained for $A=0.4$. []{data-label="angularmomentumcurve"}](sangular "fig:"){width="45.00000%"}

It should be emphasized that the crucial difference between azimuthon dynamics in circular and square waveguide originates from the fact that angular momentum of the beam is conserved in the circular waveguide, whereas it changes considerably in the square waveguide. In the right panel of Fig. \[angularmomentumcurve\] we display the angular momentum of the beam in the square waveguide for four different values 