Continuously self-focusing and continuously self-defocusing two-dimensional beams in dissipative media

Using the Lagrangian formalism, with a simple trial function for dissipative optical two-dimensional (2D) soliton beams, we show that there are two disjoint sets of stationary soliton solutions of the complex cubic-quintic Ginzburg-Landau equation, with concave and convex phase profiles, respectively. These correspond to continuously self-focusing and continuously self-defocusing types of 2D solitons. Their characteristics are distinctly different, as the energy for their existence can be generated either at the center or in the outer layers of the soliton beam. These predictions are corroborated with direct numerical simulations of the Ginzburg-Landau equation. Regions of existence in the parameter space of these two types of solutions are found and they are in reasonable agreement with the predictions of the Lagrangian approach. In addition, direct numerical simulations allow us to find more complicated localized solutions around these regions. These solutions lack cylindrical symmetry and/or pulsate in time. Examples of the complex behavior of these beams are presented.