### abstract

- We apply an approach based on the Fokker –Planck equation to study the statistics of optical soliton parameters in the presence of additive noise. This rigorous method not only allows us to reproduce and justify the classical Gordon – Haus formula but also leads to new exact results. © 2003 Optical Society of America OCIS codes: 060.2330, 190.5530. Since the seminal work by Gordon and Haus 1 (see also Ref. 2 for the mathematical theory of the effect), a great deal of attention has been devoted to studying the statistics of the soliton parameters in a long-haul optical communication. Amplified spontaneous emission yields spurious jitter in the soliton parameters, which in turn impairs transmission. The effect is enhanced as the propagation distance increases. For instance, the celebrated Gordon–Haus result for the pure nonlinear Schrodinger equation with lumped amplifiers says that the variance of the time jitter grows proportionally to the cube of the propagation distance. Most results concerning soliton jitter have been obtained under the assumption that the statistics is Gaussian. Although the results based on this assumption seem to agree rather well with numerics and experiments, from the theoretical view point a more rigorous justification is desirable. Actually, there is no reason to assume that Gaussian statistics holds for large propagation distances, because the system under consideration is essentially nonlinear. Knowledge of the correct probability density function (PDF) is especially important because such characteristics as bit-error rate depend on the shape of the whole PDF, in particular, on its tails. Large, rare f luctuations in a nonlinear system are typically beyond the area of applicability of the usual Gaussian statistics. 3–1 0 Therefore knowledge of the whole PDF, including tails, is absolutely crucial for a correct estimate of the bit-error rate. And, as we show below, these tails are not Gaussian even for small propagation distances, when the bulk of the PDF can still be considered as such. In this Letter we study soliton statistics, rigorously deriving the Fokker–Planck equation governing the PDF for the four soliton parameters. As the principal example we examine the nonlinear Schrodinger equation with additive white Gaussian noise. First we show that the Gaussian statistics can be justified for short enough propagation distances (apart from the tails). For large distances the Gaussian approximation breaks down, and we illustrate this by calculating explicitly the PDF of the soliton amplitude. In the presence of amplified spontaneous emission noise the propagation of solitons in optical fibers is described by the perturbed nonlinear Schrodinger equation: ≠q ≠z i