Discrete nonlinear Fourier transforms and their inverses. (1st August 2022)

• Record Type: Journal Article
• Authors: Saksida, Pavle
• Is Part Of: Inverse problems. Volume 38:Number 8(2022)
• Publication Date: 2022-08-01
• Subjects: nonlinear Fourier transform -- inverse nonlinear Fourier transform -- discrete nonlinear Fourier transform -- distributions -- delta functions -- inverse scattering transform; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/ac73ae
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We study two discretisations of the nonlinear Fourier transform of AKNS–ZS type, F E and F D . Transformation F D is suitable for studying the distributions of the form u = ∑ n = 1 N u n δ x n, where δ x n are delta functions. The poles x n are not equidistant. The central result of the paper is the construction of recursive algorithms for inverses of these two transformations. The algorithm for ( F D ) − 1 is numerically more demanding than that for ( F E ) − 1 . We describe an important symmetry property of F D . It enables the reduction of the nonlinear Fourier analysis of the constant mass distributions u = ∑ n = 1 N u c δ x n for the numerically more efficient F E and its inverse.