Introducing the fast nonlinear Fourier transform

The nonlinear Fourier transform (NFT; also: direct scattering transform) is discussed with respect to the focusing nonlinear Schrödinger equation on the infinite line. It is shown that many of the current algorithms for numerical computation of the NFT can be interpreted in a polynomial framework. Finding the continuous spectrum corresponds to polynomial multipoint evaluation in this framework, while finding the discrete eigenvalues corresponds to polynomial root finding. Fast polynomial arithmetic is used in order to derive algorithms that are about an order of magnitude faster than current implementations. In particular, an N sample discretization of the continuous spectrum can be computed with only O(N log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N) flops. A finite eigenproblem for the discrete eigenvalues that can be solved in O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) is also presented. The feasibility of this approach is demonstrated in a numerical example.