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(KN+C_pNlog²N) such that the error vanishes as mathop O(N^-p) where p ϵ {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=p³) and the implicit Adams method (C_p=(p-1³,p>1) of which the latter proves to be the most accurate family of methods for fast NFT.