2025 , Volume 30, № 4, p.90-107

The inverse scattering transform (IST) method allows integrating the nonlinear Schr¨odinger equation (NLSE), thereby obtaining information about the structure of the solution and the presence of solitons. NLSE is widely used, in particular, in nonlinear fiber optics, since it describes the pulse propagation in optical fiber. The paper presents a numerical method for solving the inverse Zakharov–Shabat (ZS) problem, which is a part of the IST method for the NLSE. It is described using the Gelfand–Levitan Marchenko system of equations. The method allows increasing the accuracy of solving the inverse ZS problem to the ninth order (G9 scheme), while maintaining quadratic computational complexity. The method is based on the block version of the Toeplitz Inner-Bordering algorithm of Levinson’s type. To approximate integrals, the high-precision one-sided and two-sided Gregory quadrature formulas are used, while an efficient computational algorithm employs the Woodbury formula. The main objective of the paper is to study how increasing the accuracy of the inverse problem affects the accuracy of the IST method, provided that there are no known methods higher than the sixth order of approximation for the direct problem Numerical experiments have shown that solving a direct problem even with the sixth order of approximation, the G9 scheme allows to significantly increase the accuracy of the solution of the problem in the case of normal dispersion and for some examples in the case of anomalous dispersion. It is also shown that the G9 scheme can be successfully applied in the combined Darboux method. However, to fully utilize the G9 scheme in the IST method, schemes of similar accuracy for the direct ZS problem are required.The development of efficient methods for implementing the IST method will be useful for modelling soliton data transmission systems, as well as for signal processing and coding