Modulational instability induced by randomly varying coefficients for the nonlinear Schrِdinger equation

We introduce the theory of modulational instability (MI) of electromagnetic waves in optical fibers. The model at hand is the one-dimensional nonlinear Schrِdinger equation with random group velocity dispersion and random nonlinear coefficient. We compute the MI gain which reads as the Lyapunov exponent of a random linear system. We show that the distribution of the MI gain can be expressed in terms of the log-normal statistics. The heavy tail of this probability distribution function involves very different behaviors for the sample and moment MI gains. In the anomalous dispersion regime, random fluctuations of the nonlinear coefficient reduces the sample MI gain peak, although the moment MI peak is enhanced, and the unstable bandwidth is widened. Still in the anomalous dispersion regime, random fluctuations of the group velocity dispersion reduces both the sample MI gain peak and the moment MI peak. Finally, in the normal dispersion regime, randomness extends the MI domain to the whole spectrum of modulations, and increases the MI gain peak. The linear stability analysis is confirmed by numerical simulations of the full stochastic nonlinear Schrِdinger equation.