AI 中文总结
该研究提出基于切比雪夫多项式的快速八阶帕德格式，用于直接扎哈罗夫-沙巴特问题，提升了连续非线性谱计算的精度与效率。
AI 中文摘要
本研究针对直接扎哈罗夫-沙巴特散射问题，构建了快速八阶帕德格式。这些格式基于通过Magnus展开得到的八阶指数积分器。将传统快速帕德表示直接扩展至八阶情形，会导致所考虑测试中快速变体的精度不足。为克服这一困难，我们利用儒可夫斯基映射重新表述有限实区间上的谱依赖关系，并将局部分子和分母表示为切比雪夫多项式基，这使得能够通过快速乘积树构造全局转移矩阵，同时保留紧凑的多项式表示。针对带有两种色散符号的啁啾双曲正割势的数值实验表明，所提出的基于切比雪夫的快速格式大幅提升了对应直接快速变体的精度，可用于连续非线性谱的高效计算。
英文摘要
In this work, we construct fast eighth-order Pade schemes for the direct Zakharov-Shabat scattering problem. The schemes are based on an eighth-order exponential integrator obtained from the Magnus expansion. A direct extension of the conventional fast Pade representation to the eighth-order case leads to insufficiently accurate fast variants in the considered tests. To overcome this difficulty, we reformulate the spectral dependence on a finite real interval using the Joukowski mapping and represent the local numerators and denominators in the Chebyshev polynomial basis. This makes it possible to construct the global transition matrix by a fast product tree while retaining a compact polynomial representation. Numerical experiments for chirped hyperbolic secant potentials with both signs of dispersion show that the proposed Chebyshev-based fast schemes substantially improve the accuracy of the corresponding direct fast variants and can be used for efficient computation of the continuous nonlinear spectrum.
