Delft Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherlands.
Phys Rev E. 2017 Dec;96(6-1):063302. doi: 10.1103/PhysRevE.96.063302. Epub 2017 Dec 4.
This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU(2) nonlinear Fourier transformation (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transformation are quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present a unified framework for the forward and inverse NFT using exponential one-step methods which are amenable to FFT-based fast polynomial arithmetic. Within this discrete framework, we propose a fast Darboux transformation (FDT) algorithm having an operational complexity of O(KN+Nlog^{2}N) such that the error in the computed N-samples of the K-soliton vanishes as O(N^{-p}) where p is the order of convergence of the underlying one-step method. For fixed N, this algorithm outperforms the classical DT (CDT) algorithm which has a complexity of O(K^{2}N). We further present an extension of these algorithms to the general version of DT which allows one to add solitons to arbitrary profiles that are admissible as scattering potentials in the ZS problem. The general CDT and FDT algorithms have the same operational complexity as that of the K-soliton case and the order of convergence matches that of the underlying one-step method. A comparative study of these algorithms is presented through exhaustive numerical tests.
本文考虑了非厄米 Zakharov-Shabat (ZS) 散射问题，该问题构成了定义 SU(2) 非线性傅里叶变换 (NFT) 的基础。在 Ablowitz-Kaup-Newell-Segur 形式主义中，这种传统傅里叶变换的推广的理论基础已经相当完善；然而，仍然缺乏可用于实际应用的有效数值算法。在本文中，我们提出了一种使用指数一步方法的正向和逆向 NFT 的统一框架，该方法适用于基于 FFT 的快速多项式算法。在这个离散框架内，我们提出了一种快速 Darboux 变换 (FDT) 算法，其运算复杂度为 O(KN+Nlog^{2}N)，使得计算的 K-孤子的 N 个样本的误差随着 O(N^{-p}) 消失，其中 p 是基础一步方法的收敛阶。对于固定的 N，该算法优于具有复杂度 O(K^{2}N)的经典 DT (CDT) 算法。我们进一步将这些算法扩展到一般版本的 DT，允许将孤子添加到任意作为 ZS 问题散射势的可接受的轮廓中。一般的 CDT 和 FDT 算法具有与 K-孤子情况相同的运算复杂度，收敛阶与基础一步方法匹配。通过详尽的数值测试，对这些算法进行了比较研究。
