### Lukinov V.L.

## Statistical modeling of the dynamics of interaction of solitons in optical fiber networks

All the time need to increase the speed of data transmission is the main reason for the modern revolutionary development of communication networks, which explains the real motivation of all optical communications research. To ensure the exponential growth of data traffic, such high-tech technological solutions as wavelength division multiplexing (WDM), erbium-doped amplifiers, forward error correction, and Raman amplification were developed. At present, the theoretical limit of bandwidth of standard single-mode fiber (SSFM) has been achieved, which is due to the effects of non-linearity of fibers (Kerr effect) [1]. Intensively developed methods of suppressing the Kerr effect, associated with nonlinear compensation, have not overcome many difficulties, mainly because the existing bandwidth enhancing technologies have been developed for linear communication channels. The problem of finding the true limits of non-linear traffic channels has not yet been solved.

This paper is devoted to modeling the propagation and interaction of optical signals in fiber optic channels, which is accurately described by the nonlinear Schrödinger equation (NLSE) [1]. It is well known that NLSE (without perturbation) belongs to the class of integrable nonlinear systems. In particular, this means that NLSE admits the existence of a special type of solutions: highly robust non-linear waves, called solitons [6,7]. Solitons were proposed as the information carriers for the high-capacity fiber-optic communications.

The numerical solution of the nonlinear Schrödinger equation was carried out by applying nonlinear analogs of direct (FNFT) and inverse (BNFT) Fourier transform [2]. At FNFT, the spectrum of the incoming signal is decomposed by solving the Zakharov-Shabat equation [3]. The propagation of the nonlinear part of the spectrum is described by a known equation [1]. To find the received signal at BNFT, Monte-Carlo methods developed by the author are used to numerically solve the Gelfand-Levitan-Marchenko integral equation [4, 5, 6, 7].

Numerically modelling was conduct for investigation the power of soliton pulses at distances of 6–10 km during the passage of a random heterogeneous medium and high-frequency interactions of solitons arising at maximum loads of an optical fiber.

This work was financially supported by the Russian Foundation for Basic Research (project code 17-01-00698).

References

1. G. Agrawal, Nonlinear fiber optics, Academic Press, New York, 1996

2. S.T. Lee, J.E. Prilepsky, S.K. Turitsyn, Nonlinear inverse synthesis for high spectral efficiency transmission in optical fibers. OPTICS EXPRESS, Vol. 22, No 22, 2014.

3. V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media, // Sov. Phys. 34, 6269, 1972.

4. M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53, 249315, 1974.

5. V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Scattering Method. Colsultants Bureau, New York, 1984.

6. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, 1981.

7. A. C. Newell, Solitons in mathematics and physics. SIAM, Philadelphia, 1985.

8. T.A. Averina and S.S Artemiev, Analysis of Accuracy of Monte Carlo Methods in Solving Boundary Value Problems by Probabilistic Representations, // Sib. Zh. Vych. Mat., (2008), vol. 11, no. 3, pp. 239-250.

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