Linear modulational stability analysis of Ginzburg–Landau dissipative vortices

Two-dimensional dissipative solitons are described by the complex Ginzburg–Landau equation, with cubic-quintic nonlinearity compensating for diffraction, while linear and nonlinear losses are simultaneously balanced by the gain. Vortices with zero electric field in the center, corresponding to a topological singularity, are particularly sensitive to the azimuthal modulational instability that causes filamentation for some values of dissipative parameters. We perform linear stability analysis, in order to determine for which values of parameters the dissipative vortex either splits into filaments or becomes stable dissipative vortex soliton. The growth rates of different modulational instability modes is established. In the domain of dissipative parameters corresponding to the zero maximal growth rate, steady state solutions are stable. Analytical results are confirmed by numerical simulations of the full complex radially asymmetric cubic-quintic Ginzburg–Landau equation.