Stability of three-dimensional self-trapped beams with a dark spot surrounded by bright rings of varying intensity.

We analytically and numerically examine the stability of three-dimensional self-trapped beams with a dark spot surrounded by bright rings of varying intensity in a uniform saturable self-focusing medium. It is shown that the fundamental bound state of the family is stable to a symmetric perturbation but unstable to an asymmetric perturbation (that breaks the azimuthal symmetry of the beam, i.e. , transverse instabilities). The higher-order states are also found to display transverse (modulation) instabilities. The development of the instabilities is shown to lead to the emission of filaments which spiral away from the center of the dark spot as stable entities.