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 \mathop {O}(KN+C_{p}N\log ^{2}N) such that the error vanishes as \mathop {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},\,p>1 ) of which the latter proves to be the most accurate family of methods for fast NFT.