Modulational instability in optical fiber with stochastic parameters and noninstantaneous response

We investigate analytically and numerically the modulational instability (MI) in optical fiber, where the effect of noninstantaneous nonlinear response as well as stochastic coefficients are taken into account. Applying the linear stability analysis and stochastic calculus, we show that the MI gain spectrum reads as the maximal eigenvalue of a constant matrix. In the limiting cases of small fluctuations, we give explicit expressions for the MI gain spectra. In the general configurations, we derive an explicit form of the effective matrix and compute numerically the maximal eigenvalue. The moment MI peak is enhanced and the delayed Raman response reduces the maximum MI gain caused by stochasticity both in anomalous and normal dispersion regimes. Numerical simulations of the full stochastic nonlinear Schródinger equation show that, the phenomenon of MI gives rise to periodic pulse arrays of waves train, as well as to a chain of peaks with continuously growing amplitudes.