We describe in detail the construction of numerical fourth- and sixth-order schemes using the Magnus expansion for solving the Zakharov–Shabat system, which makes it possible to solve accurately the direct scattering problem for the nonlinear Schrödinger equation. To avoid numerical instabilities inherent in the procedure of solving the direct scattering problem, high-precision arithmetic is used. At present, application of the proposed schemes in combination with the highprecision arithmetic is a unique tool that can be used for analysis of complex wave fields, which contain a great number of solitons, and allows one to determine the complete discrete spectrum, including both eigenvalues and normalization constants. In this work, we study the errors in the proposed scheme using an example of the potential in the form of a hyperbolic secant. It is found that the time of calculation of the scattering matrix using the sixth-order algorithm is almost twice as long compared with that for the standard second-order Boffetta–Osborne algorithm, whereas the time gain resulting from the reduction of number of the wave-field discretization points with retaining the desired precision can reach an order of magnitude or more. Exact solution of the direct scattering problem requires that the discretization increment of high-amplitude wave fields should be comparable with the characteristic width of the largest solitons contained in such fields. In this case, the discretization increment can be sufficiently smaller than that required for reconstruction of the full Fourier spectrum of the wave field. Application of the proposed high-order approximation schemes can be of fundamental importance for successful operation with a great number of complex nonlinear wave fields, such as, e.g., that in the process of the statistical study of the scattering data.

我们详细描述了使用马格努斯展开式求解 Zakharov-Shabat 系统的数值四阶和六阶方案的构造，这使得准确求解非线性薛定谔方程的直接散射问题成为可能。为了避免解决直接散射问题的过程中固有的数值不稳定性，使用了高精度算法。目前，所提出的方案与高精度算法相结合的应用是一种独特的工具，可用于分析包含大量孤子的复杂波场，并可以确定完整的离散谱，包括两个特征值和归一化常数。在这项工作中，我们使用双曲正割形式的势示例来研究所提出方案中的误差。发现使用六阶算法计算散射矩阵的时间几乎是标准二阶 Boffetta-Osborne 算法的两倍，而波数减少导致的时间增益保持所需精度的场离散点可以达到一个数量级或更多。直接散射问题的精确解要求高振幅波场的离散化增量应与此类场中包含的最大孤子的特征宽度相当。在这种情况下，离散化增量可以比重构波场的完整傅立叶谱所需的增量小得多。