of modulation parameter $B$. If $B$ is above the critical value $B_{\rm cr}$, the angular momentum does not change its sign as the blue curve shows. If $B<B_{\rm cr}$, the angular momentum changes periodically its sign in propagation (red curve), which indicates the swinging motion of the soliton. For values $B$ close to the critical value the variation of the angular momentum are largest as shown by the green and black curves. At the same time, the appearance of broad plateaus indicates slow propagation dynamics.

It is worth mentioning that we also investigated other waveguide symmetries, such as hexagonal waveguides, and found the behavior of the dipole azimuthons to be qualitatively similar to that of the square waveguide, i.e., they either rotate or wobble, depending on modulation depth and initial orientation. However, the difference of the values of the Hamiltonian for the two extremal dipoles is now smaller, resulting in a lower threshold value $B_{\rm cr}$. We believe that this threshold behavior in the propagation dynamics of azimuthons is generic for any non-circular-symmetric waveguide structure which supports degenerated dipole modes.

Rotating higher order localized modes {#higherorder}
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In analogy to dipole-azimuthons discussed so far, it is possible to consider dynamics 