Deterministic effective equations for the propagation of expectation in noisy nonlinear optical fibers

We consider electromagnetic propagation in optical fibers as described by a nonlinear Schroedinger equation. The equation is endowed with input data representing a signal affected by a Gaussian white noise. Assuming both the nonlinearity and the noise to be small of the same order epsilon, we derive a hierarchy, at various orders in epsilon, of deterministic effective equations for the expectation of the propagated signal.