A parametric nonlinear Schrödinger equation and stability criterion

Stability investigation is carried out for a parametric nonlinear Schrödinger equation containing the first order spatial and the first order temporal derivatives of a complex conjugate type. It is derived formally from an arbitrary system in which a scalar dependent variable u satisfies an equation of the form (L + α)u = N. L is a linear operator involving the temporal and spatial differential operators, α is a periodic function of time and N represents all the nonlinear terms. A useful analysis of dispersive perturbations for a periodic temporal solution is developed. A stability criterion for arbitrary linear disturbance is obtained.