Stable vortex tori in the three-dimensional cubic-quintic Ginzburg-Landau equation

We demonstrate the existence of stable toroidal dissipative solitons with the inner phase field in the form of rotating spirals, corresponding to vorticity S=0, 1, and 2, in the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity. The stable solitons easily self-trap from pulses with embedded vorticity. The stability is corroborated by accurate computation of growth rates for perturbation eigenmodes. The results provide the first example of stable vortex tori in a 3D dissipative medium, as well as the first example of higher-order tori (with S=2) in any nonlinear medium. It is found that all stable vortical solitons coexist in a large domain of the parameter space; in smaller regions, there coexist stable solitons with either S=0 and S=1, or S=1 and S=2.