Stabilization mechanism for two-dimensional solitons in nonlinear parametric resonance

We consider a simple model system supporting stable solitons in two dimensions. The system is the parametrically driven damped nonlinear Schrodinger equation, and the soliton stabilizes for sufficiently strong damping. We elucidate the stabilization mechanism by reducing the partial differential equation to a, finite-dimensional dynamical system and conclude that the negative feedback loop occurs via enslaving the soliton phase, locked to the driver, to its amplitude and width.