Discrete nonlinear Fourier transforms and their inverses

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.