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 complexity of O(K N t C
_p
N log
²
N) such that the error vanishes as O(N
^-p
) where p ∈ {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
, p > 1) of which the latter proves to be the most accurate family of methods for fast NFT.