We study a stochastic Nonlinear Schrödinger Equation (NLSE), with additive white Gaussian noise, by means of the Nonlinear Fourier Transform (NFT). In particular, we focus on the propagation of discrete eigenvalues along a focusing fiber. Since the stochastic NLSE is not exactly integrable by means of the NFT, then we use a perturbation approach, where we assume that the signal-to-noise ratio is high. The zeroth-order perturbation leads to the deterministic NLSE while the first-order perturbation allows to describe the statistics of the discrete eigenvalues. This is important to understand the properties of the channel for recently devised optical transmission techniques, where the information is encoded in the nonlinear Fourier spectrum.

In the present paper we propose four systems of linear Partial Di¤erential Equations that can be deduced from the nonlinear Schro ¨dinger equation for the propagation of light in optical fibers in the frame of the recently-proposed Combined Regular-Logarithmic Perturbation method. The unknown function in the Schro ¨dinger equation is the optical field envelope; it is a complexvalued function. Following the Combined Regular-Logarithmic Perturbation method, proposed by Secondini, Forestieri and Menyuk, we look for complex solutions of the Schro ¨dinger equation in the form of a perturbed continuous wave that relates three unknown real-valued functions. Since the Schro ¨dinger equation is complex, we split it into two real equations, both in the three real unknowns. We linearize these two equations and add a third linear equation that relates the same three unknown quantities. We propose four di¤erent choices for the third equation, therefore we obtain four di¤erent real systems of linear Partial Di¤erential Equations and we analyze the corresponding systems of Ordinary Di¤erential Equations for the Fourier transforms of the unknowns. One of the four systems we obtain is equivalent to that studied by the quoted authors. We add to it other three choices that could be useful to model di¤erent situations. Again, we consider the real part of the Ordinary Di¤erential Equations and we present solutions in recursive form. We also suggest solutions for the complex-valued Fourier transforms by using Bessel functions.

No abstract