Higher Order Convergent Fast Nonlinear Fourier Transform

Abstract

It is demonstrated is 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 complexity of $O{KN+C_pN\log^2N}$ such that the error vanishes as $O{N^{-p}}$ where $p\in\{1,2,3,4\}$ and $K$ is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1)^3$) of which the latter proves to be the most accurate family of methods for fast NFT.