  The achievable data rates of current fiber-optic
wavelength-division-multiplexing (WDM) systems are limited by nonlinear
interactions between different subchannels. Recently, it was thus proposed to
replace the conventional Fourier transform in WDM systems with an appropriately
defined nonlinear Fourier transform (NFT). The computational complexity of NFTs
is a topic of current research. In this paper, a fast inverse NFT algorithm for
the important special case of multi-solitonic signals is presented. The
algorithm requires only $\mathcal{O}(D\log^{2}D)$ floating point operations to
compute $D$ samples of a multi-soliton. To the best of our knowledge, this is
the first algorithm for this problem with $\log^{2}$-linear complexity. The
paper also includes a many samples analysis of the generated nonlinear Fourier
spectra.
