A higher-order perturbation analysis of the nonlinear Schrödinger equation

•Higher-order perturbation analysis of the nonlinear Schrödinger equation.•Explicit formulas for the propagation of the mean spectral density of white noise as a perturbation to a continuous-wave pump in a nonlinear optical fiber.•Excellent agreement with numerical simulations and experimental measurements.•The proposed formulas give some insights into the physics behind the experimentally observed behavior. A well-known and thoroughly studied phenomenon in nonlinear wave propagation is that of modulation instability (MI). MI is usually approached as a perturbation to a pump, and its analysis is based on preserving only terms which are linear on the perturbation, discarding those of higher order. In this sense, the linear MI analysis is relevant to the understanding of the onset of many other nonlinear phenomena, such as supercontinuum generation, but it has limitations as it can only be applied to the propagation of the perturbation over short distances. In this work, we propose approximations to the propagation of a perturbation, consisting of additive white noise, that go beyond the linear modulation instability analysis, and show them to be in excellent agreement with numerical simulations and experimental measurements.