Fast sixth-order algorithm based on the generalized Cayley transform for the Zakharov-Shabat system associated with nonlinear Schrodinger equation

The nonlinear Schrodinger equation (NLSE) is widely used in telecommunication applica-tions, since it allows one to describe the propagation of pulses in an optical fiber. Recently some new approaches based on the nonlinear Fourier transform (NFT) have been actively explored to compensate for fiber nonlinearity and to exceed the limitations of nonlinearity-imposed limits of linear transmission methods. Despite the fact that the numerical solution of NLSE is a general problem, nevertheless, the optical community has been focusing on this issue. Improving the accuracy of the NFT algorithms remains an urgent problem in optics. In particular, it is important to increase the approximation order of the methods, especially in problems where it is necessary to analyze the structure of complex wave-forms. To correctly describe them and their spectral parameters, more accurate and fast numerical methods are needed. We propose a novel general approach for constructing sixth-order (with respect to an in-tegration step) finite-difference schemes for first-order linear differential systems. These schemes are based on the generalized Cayley transform and include exponential integra-tors as a special case. If the system has a time-dependent skew-hermitian matrix then the schemes conserve the quadratic first integral automatically. Then we apply our method to solve the direct spectral problem for the Zakharov-Shabat system. New schemes with frac-tional rational transition matrix allow the use of fast algorithms to solve the initial problem for a large number of values of the spectral parameter. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

NLSE is commonly used in telecommunications to describe pulse propagation in optical fibers. Recent research has been focused on improving NFT algorithms in optics, particularly in increasing approximation order to analyze complex waveforms.NewProposed schemes for first-order linear differential systems are based on the generalized Cayley transform and can automatically conserve quadratic first integrals in systems with time-dependent skew-hermitian matrices.