Modulation instability analysis of a nonautonomous $$(3+1)$$-dimensional coupled nonlinear Schrödinger equation

Abstract

We have investigated the modulation instability (MI) analysis of a nonautonomous \((3+1)\)-dimensional coupled nonlinear Schrödinger (NLS) equation with time-dependent dispersion and phase modulation coefficients. By employing standard linear stability analysis, we have obtained an explicit expression for the MI gain as a function of dispersion, phase modulation, perturbation wave numbers and an initial incidence power. A nonautonomous coupled NLS equation is found to be modulationally unstable for the same sign of dispersion and phase modulation coefficients. This equation is modulationally stable for zero dispersion and or phase modulation. For nonzero dispersion, the equation is found to be modulationally stable/unstable on distinct bandwidth of wave numbers. The trigonometric, exponential, algebraic functions of time and constant have been chosen as test functions for dispersion and phase modulation to study the effect on the MI analysis. The effect of focusing and defocusing medium on the MI analysis has also been investigated. The MI bandwidth in the focusing medium is found to be larger than defocusing medium. It is found that the MI of the equation can be managed by proper choice of the dispersion and phase modulation parameters.