Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schr'odinger equation

摘要

Starting from a comparison of some established numerical algorithms for thecomputation of the eigenvalues (discrete or solitonic spectrum) of thenon-Hermitian version of the Zakharov-Shabat spectral problem, this articledelivers new algorithms that combine the best features of the existing ones andthereby allays their relative weaknesses. Our algorithm is modelled within theremit of the so-called direct nonlinear Fourier transform (NFT) associated withthe focusing nonlinear Schr\"odinger equation. First, we present the data forthe calibration of methods comparing the relative errors associated with thecomputation of the continuous NF spectrum. Then each method is paired withdifferent numerical algorithms for finding zeros of a complex-valued functionto obtain the eigenvalues. Next we describe a new class of methods based on thecontour integrals evaluation for the efficient search of eigenvalues. Afterthat, we introduce a new hybrid method, one of our main results: the methodcombines the advances of contour integral approach and makes use of theiterative algorithms at its second stage for the refined eigenvalues search.The veracity of our new hybrid algorithm is established by estimating theconvergence speed and accuracy across three independent test profiles. Alongwith the development of a new approach for the computation of the eigenvalues,our study also addresses the problem of computation of the so-called normingconstants associated with the eigenvalues. We show that our formalismeffectively amounts to accurate and fast enough computation of residues of thereflection coefficient in the upper complex half-plane of the spectralparameter.