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 [1-4]. In the presence of high order nonlinear responses, these equations exhibit instabilities that lead to wave collapse [1, 4]. 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 [4]. 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 [4], 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 [1,3-4].