Fast inverse nonlinear Fourier transform

This paper considers the non-Hermitian Zakharov-Shabat scattering problem which forms the basis for defining the $\mathrm{SU}\left(2\right)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is 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 two fast inverse NFT algorithms with $O(KN+N{log}^{2}N)$ complexity and a convergence rate of $O\left(N^{-2}\right)$, where $N$ is the number of samples of the signal and $K$ is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme $\left[O\left(N{log}^{2}N\right)\right]$ together with a new fast Darboux transformation (FDT) algorithm $\left[O(KN+N{log}^{2}N)\right]$ previously developed by V. Vaibhav [Phys. Rev. E 96, 063302 (2017)]. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

Figure 1

The figure shows a schematic of the fast inverse NFT (INFT) algorithm where the dashed line depicts the missing part of the algorithm to be discussed in this article (the FDT algorithm has been reported in Ref. [3]). Here, $q_R\left(t\right)$ refers to the “radiative” part of the signal $q\left(t\right)$ which is obtained as a result of removing the bound states. Note that $q_R\left(t\right)$ has the reflection coefficient ${\rho }_R\left(\xi \right)$ [see Sec. 2 for the connection between ${\rho }_R\left(\xi \right)$ and $\rho \left(\xi \right)]$.

Figure 2

The figure depicts the equivalent layered-media for the discrete scattering problem in Sec. 3. In each of the layers, the ZS problem is approximated by two instantaneous scatterers and a “free-space” propagation between them.

Figure 3

The figure shows a comparison of the algorithms LP and TIB for the secant-hyperbolic potential ($A_R=0.4$) with respect to convergence rate (left) and run time per sample (right).

Figure 4

The figure shows the performance of the algorithms INFT-A, -B, and -C for a fixed number of eigenvalues ($K\in \{12,16,20\}$) and varying number of samples ($N$) for the secant-hyperbolic potential (see Sec. 4a). The error plotted on the vertical axis is defined by (26).

Figure 5

The figure shows the performance of the algorithms INFT-A, -B, -C for a fixed number of samples ($N\in \{2^{13},2^{14},2^{15}\}$) and varying number of eigenvalues ($K$) for the secant-hyperbolic potential (see Sec. 4a). The error is quantified by (26).

Figure 6

The figure shows the error analysis for the signal generated from the continuous spectrum given by (28) which is the frequency-domain description of the raised-cosine filter (see Sec. 4b). The error is quantified by (31).

Figure 7

The figure shows the potential corresponding to a QPSK modulated continuous spectrum given by (32) with number of symbols $N_{\text{sym}}\in \{16,32\}$. The number of samples used is $N=2^{12}$ and the computational domain is $[-15T_2,T_2]$, where $T_2$ is given by (33). Also, we set $A_{\text{eff}}=10$ which is defined by (34).

Figure 8

The figure shows the error analysis for the QPSK modulated continuous spectrum given by (32) for varying number of symbols $N_{\text{sym}}$ (see Sec. 4b). Here we set $A_{\text{eff}}=10$, which is defined by (34).

Figure 9

The figure shows the results of the error analysis for an example where the discrete spectrum is ${\mathfrak{S}}_K$ and the continuous spectrum is identical to the Fourier spectrum of the raised-cosine filter (see Sec. 4b1). The error plotted on the vertical axis corresponds to the continuous spectrum, which is quantified by (31). Here $A_{\text{rc}}=20$.

Figure 10

The figures shows the results of the error analysis for an example where the discrete spectrum is ${\mathfrak{S}}_K$ and the continuous spectrum is identical to the Fourier spectrum of the raised-cosine filter (see Sec. 4b). The error plotted on the vertical axis corresponds to the norming constant, which is quantified by (35). Here $A_{\text{rc}}=20$.

Figure 11

The figure shows a comparison of convergence behavior and run time of NFT algorithms based on the discretization schemes, namely ${\mathrm{IA}}_m$ ($m\in \{1,2,3\}$) and the Split-Magnus (SM) and Magnus methods with one-point Gauss quadrature (MG1) (the latter two are discussed in Ref. [3, Sec. 4] as a way of benchmarking). The method ${\mathrm{IA}}_1$ is identical to the trapezoidal rule. The test corresponds to a secant-hyperbolic profile $q\left(t\right)=4.4sech\left(t\right)$.