Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem

Abstract

We propose a new high-precision algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and is a generalization of the second order Boffetta-Osborne scheme. It is allowed by our method to solve more effectively the Zakharov-Shabat spectral problem for continuous and discrete spectra. Keywords Zakharov-Shabat problem, inverse scattering transform, nonlinear Schrödinger equation, numerical methods The solution of the direct problem for the Zakharov-Shabat problem (ZSP) is the first step in the inverse scattering transform (IST) for solving the nonlinear Schrödinger equation (NLSE) [1]. The numerical implementation of the IST has gained great importance and attracted special attention since Hasegawa and Tappert [2] proposed to use soliton solutions as a bit of information for fiber optic data transmission.

Nonlinear Spectrum of Conventional OFDM and WDM Return-to-Zero Signals in Nonlinear Channel

“…Considered NFT processing of signals require efficient numerical algorithms such as, e.g. [27]- [29]. These methods will be useful for application of NFT in other areas beyond optical communications.…”

Direct scattering transform: catch soliton if you can

“…1. We should stress that both formulas (11) and ( 12) are verified numerically in the ζ-plane, see Supplementary materials. However, concerning the accuracy of b N,tr close to the real axis a small deviation can still arise due to the fact that the leading order term and o N become of the same order.…”

“…After several decades of analytical studies of integrable equations, the rapid growth of interest to describe arbitrary shaped, noisy and even random nonlinear wave fields has promoted the need in accurate numerical methods for the DST. The Boffetta-Osborne method [7] represents the first numerical realization of the DST followed by a sequence of further improvements and alternatives [8][9][10][11][12]. These advancements have made the DST an essential scientific tool with a wide range of theoretical and experimental applications [4,[13][14][15][16][17].…”

Full-Spectrum Periodic Nonlinear Fourier Transform Optical Communication Through Solving the Riemann-Hilbert Problem

“…We also point out several possible ways of how to improve the performance of PNFT-based systems further, mentioning, in particular, the more involved structure of main spectrum, the use of 2π range for the phase modulation, the optimisation of sampling in both time and nonlinear spectral domains, and the utilisation of, e.g., pilots to get rid of deterministic offsets. Some other advancements may also include, e.g., the incorporation of improved Zakharov-Shabat solution methods at the receiver [44,45], or some improved methods for solving the RHP numerically. Overall, based on the results obtained, we can assert that the proposed approach, though being so far in its rather nascent stage, allows us to gain a great flexibility, rectifying some drawbacks of "conventional" NFT systems and admitting further modification/improvements: it already reveals relatively deserving performance metrics, thus demonstrating its potential in achieving the high efficiency in optical communication systems' functioning.…”