Phase computation for the finite-genus solutions to the focusing nonlinear Schrödinger equation using convolutional neural networks

We develop a method for retrieving a set of parameters of a quasi-periodic finite-genus (finite-gap) solution to the focusing nonlinear Schrödinger (NLS) equation, given the solution evaluated on a finite spatial interval for a fixed time. These parameters (named “phases”) enter the jump matrices in the Riemann-Hilbert (RH) problem representation of finite-genus solutions. First, we detail the existing theory for retrieving the phases for periodic finite-genus solutions. Then, we introduce our method applicable to the quasi-periodic solutions. The method is based on utilizing convolutional neural networks optimized by means of the Bayesian optimization technique to identify the best set of network hyperparameters. To train the neural network, we use the discrete datasets obtained in an inverse manner: for a fixed main spectrum (the endpoints of arcs constituting the contour for the associated RH problem) and a random set of modal phases, we generate the corresponding discretized profile in space via the solution of the RH problem, and these resulting pairs – the phase set and the corresponding discretized solution in a finite interval of space domain – are then employed in training. The method's functionality is then verified on an independent dataset, demonstrating our method's satisfactory performance and generalization ability

Capacity estimates for optical transmission based on the nonlinear Fourier transform

What is the maximum rate at which information can be transmitted error-free in fibre-optic communication systems? For linear channels, this was established in classic works of Nyquist and Shannon. However, despite the immense practical importance of fibre-optic communications providing for >99% of global data traffic, the channel capacity of optical links remains unknown due to the complexity introduced by fibre nonlinearity. Recently, there has been a flurry of studies examining an expected cap that nonlinearity puts on the information-carrying capacity of fibre-optic systems. Mastering the nonlinear channels requires paradigm shift from current modulation, coding and transmission techniques originally developed for linear communication systems. Here we demonstrate that using the integrability of the master model and the nonlinear Fourier transform, the lower bound on the capacity per symbol can be estimated as 10.7 bits per symbol with 500 GHz bandwidth over 2,000 km