Efficient Nonlinear Fourier Transform algorithms of orderfFour on equispaced grid

We explore two classes of exponential integrators, in this letter, to design the nonlinear Fourier transform (NFT) algorithms with a convergence order of four on an equispaced grid. The integrating factor-based method in the class of the Runge-Kutta methods yields algorithms with complexity O(N\log²N) (where N is the number of...

Fast inverse nonlinear Fourier transform

This paper considers the non-Hermitian Zakharov-Shabat scattering problem which forms the basis for defining the SU(2)-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however, efficient numerical...

Higher order convergent fast nonlinear Fourier transform

It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a...

Nonlinear Fourier transform of time-limited and one-sided signals

In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform. In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal...

Exact solution of the Zakharov–Shabat scattering problem for doubly-truncated multisoliton potentials

Recent studies have revealed that multisoliton solutions of the nonlinear Schrödinger equation, as carriers of information, offer a promising solution to the problem of nonlinear signal distortions in fiber optic channels. In any nonlinear Fourier transform based transmission methodology seeking to modulate the discrete spectrum of the...

Fast inverse nonlinear Fourier transformation using exponential one-step methods: Darboux transformation

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU(2) nonlinear Fourier transformation (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transformation are quite well established in the Ablowitz-Kaup-Newell-Segur formalism; however,...