Discrete nonlinear Fourier transforms and their inverses

We study two discretisations of the nonlinear Fourier transform of AKNS-ZS type, ${\cal F}^E$ and ${\cal F}^D$. Transformation ${\cal F}^D$ is suitable for studying the distributions of the form $u = \sum_{n = 1}^N u_n \, \delta_{x_n}$, where $\delta _{x_n}$ 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 $({\cal F}^D)^{- 1}$ is numerically more demanding than that for $({\cal F}^E)^{- 1}$. We describe an important symmetry property of ${\cal F}^D$. It enables the reduction of the nonlinear Fourier analysis of the constant mass distributions $u = \sum_{n = 1}^N u_c \, \delta _{x_n}$ for the numerically more efficient ${\cal F}^E$ and its inverse.