Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation

Abstract

Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov-Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modeled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrödinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter. ∞ −∞ |q(t, z 0 )| dt < ∞.(2)

Exponential fourth order schemes for direct Zakharov-Shabat problem

“…(59) For C = 0 it is a well-known Satsuma-Yajima signal. The detailed numerical results for this potential are presented in [4].…”

“…The NLSE describes the envelope for wave beams, therefore it is used in many areas of physics where there are wave systems. Despite a large number of articles [2,3,4] devoted to NFT, the development of the accurate and fast numerical algorithms for NFT still remains an actual mathematical problem.…”

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem

“…Currently, one of the most effective methods for solving the ZSP is the Boffetta-Osborne (BO) method [6], which has the second order of approximation. Comparisons of this method with other methods were carried out in [5,7].…”

Contour integrals for numerical computation of discrete eigenvalues in the Zakharov–Shabat problem

“…The zeros of the spectral function a(ξ) in the upper half-plane of ξ represent solitonic eigenvalues, and the norming constants in case of simple zeros are the residues of r(ξ) at its poles (when b is analytic at these points). For our purposes it is important that a(ξ) is an analytic function in the upper half-plane of ξ, while b(ξ) is analytic in the region ξ > 0 only if q(τ) has a finite support [7]. In this Letter we focus on the numerical computation of discrete (solitonic) eigenvalues, i.e.…”

“…So it is possible to improve the contour integrals approach accuracy by the larger factor than grid search improvement gaining less penalty in complexity. When using the additional iterative refinement to reach the acceptable eigenvalue accuracy, as it was done in [7,9] is the number of steps to reach the desired zero with a given accuracy. But when we have an insufficiently accurate initial guess, the number of required iterative steps N iter can be dramatically large, going to infinity when the algorithm cannot converge.…”