Wave instabilities in nonlinear schrödinger systems with nonvanishing background

The generalized Nonlinear Schrödinger Equation (GNLSE): i∂φ/∂ t+1/2 Δ φ + F(| φ|²) φ = 0 is a fundamental equation for the universal propagation of dispersive and nonlinear waves. In the presence of high order nonlinear responses, these equations exhibit instabilities that lead to wave collapse. The study of collapse has stirred significant interest in scientific community, especially in Optics, as it lead to the localization and trapping of energy in small spatial scales. To date, most efforts have been directed to the study of localized pulses with vanishing boundary conditions, where collapse is demonstrated to occur when the field Hamiltonian is negative, while practically nothing is known in the presence of a nonzero background. The latter is a particularly important in Optics, due to the large interest stirred by the study of nonlinear waves with nonzero background, such as e.g., Dark/Gray solitons.