Nonlinear Schrödinger equation (NLSE) is often used as a master path-average model for fiber-optic transmission lines. In general, NLSE describes the coexistence of dispersive waves and soliton pulses. Propagation of signal in such a nonlinear channel is conceptually different from linear systems. We demonstrate here that the conventional orthogonal frequency-division multiplexing (OFDM) input optical signal at powers typical for modern communication systems might have soliton components statistically created by the random process corresponding to the information content. Applying Zakharov-Shabat spectral problem to a single OFDM symbol with multiple sub-carriers we quantify the effect of statistical soliton occurrence in such an information-bearing optical signal. Moreover, we observe that at signal powers optimal for transmission OFDM symbol incorporates multiple solitons with high probability. The considered optical communication example is relevant to a more general physical problem of generation of coherent structures from noise.

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The nonlinear Schrödinger equation (NLSE) is often used as a master path-average model for fiber-optic links to analyse fundamental properties of such nonlinear communication channels. Transmission of signal in nonlinear channels is conceptually different from linear communications. We use here the NLSE channel model to explain and illustrate some new unusual features introduced by nonlinearity. In general, NLSE describes the co-existence of dispersive (continuous) waves and localised (here in time) waves -soliton pulses. The nonlinear Fourier transform method allows one to compute for any given temporal signal the so-called nonlinear spectrum, that defines both continuous spectrum (analogue to conventional Fourier spectral presentation) and solitonic components. Nonlinear spectrum remains invariant during signal evolution in the NLSE channel. We examine conventional orthogonal frequency-division multiplexing (OFDM) and wavelength-division multiplexing (WDM) return-tozero signals and demonstrate that both signals at certain power levels have soliton component. We would like to stress that this effect is completely different from the soliton communications studied in the past. Applying Zakharov-Shabat spectral problem [1] to a single WDM or OFDM symbol with multiple sub-carriers we quantify the effect of statistical occurrence of discrete eigenvalues in such an information-bearing optical signal. Moreover, we observe that at signal powers optimal for transmission an OFDM symbol with high probability has a soliton component.

A neural network architecture is proposed that allows a continuous nonlinear spectrum of optical signals to be predicted and an inverse nonlinear Fourier transform (NFT) to be performed for signal modulation. The average value of the relative error in predicting the continuous spectrum by the neural network when calculating the direct NFT is found to be 2.68 × 10-3, and the average value of the relative error in predicting the signal for the inverse NFT is 1.62 × 10-4.

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture.