Efficient Nonlinear Fourier Transform algorithms of orderfFour on equispaced grid

We explore two classes of exponential integrators, in this letter, to design the nonlinear Fourier transform (NFT) algorithms with a convergence order of four on an equispaced grid. The integrating factor-based method in the class of the Runge-Kutta methods yields algorithms with complexity O(N\log²N) (where N is the number of samples of the signal), which have superior accuracy-complexity tradeoff than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield the algorithms of complexity O(N²) that have superior error behavior than that of the fast methods.