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 Schrödinger equation, and the soliton stabilises for sufficiently strong damping. The purpose of this note is to elucidate the stabilisation mechanism; we do this by reducing the partial differential equation to a finite-dimensional dynamical system. Our conclusion is that the negative feedback loop occurs via the enslaving of the soliton’s phase, locked to the driver, to its amplitude and width. 1. When a liquid layer is subjected to vertical vibration, a oneor twodimensional periodic pattern forms on its surface. This phenomenon has been known since the celebrated Faraday resonance experiment [1]; more recently, it was found that the vertical vibration is also capable of sustaining localised 2D states. These spatially localised, temporally oscillating structures — commonly referred to as oscillons — were observed on the surface of granular materials [2], Newtonian [3, 4] and non-Newtonian [5] fluids. Subsequently, stable oscillons were reproduced in numerical simulations within a variety of models, including the order-parameter equations [6, 4], discrete-time maps with continuous spatial coupling [7], semicontinuum [8] and hydrodynamic [9] theories. Although these simulations accounted for the formation of oscillons in several particular physical settings, they did not uncover the core of the mechanism which makes them immune from the nonlinear blow-up and dispersive broadening. The fact that stable oscillons occur in diverse physical media and in mathematical models of various nature, suggests that this mechanism is simple and general. It should operate whenever one has