Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media

We present a detailed stability analysis of one- and two-ring solitary waves with central phase dislocation in self-focusing saturable and quadratic nonlinear media. Varying parameters, we demonstrate transitions between different filamentation scenarios. An analytical approach is developed for the study of filament dynamics after ring breakup. Approximate expressions for the angular separation rate of filaments based on the conservation of the angular momentum and on the conservation of the Hamiltonian are derived and compared. The stability analysis and analytical results are tested by an extensive series of numerical simulations of the original models, and good agreement is found.