  We explore two classes of exponential integrators in this letter to design
nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity
trade-off and a convergence order of $4$ on an equispaced grid. The integrating
factor based method in the class of Runge-Kutta methods yield algorithms with
complexity $O(N\log^2N)$ (where $N$ is the number of samples of the signal)
which have superior accuracy-complexity trade-off than any of the fast methods
known currently. The integrators based on Magnus series expansion, namely,
standard and commutator-free Magnus methods yield algorithms of complexity
$O(N^2)$ that have superior error behavior even for moderately small step-sizes
and higher signal strengths.
