On the rigorous justification of b-modulation method and inclusion of discrete eigenvalues

Addressing the optical communication systems
employing the nonlinear Fourier transform (NFT)
for the data modulation/demodulation, we provide the
explicit proof for the properties of the signals emerging
in the so-called b-modulation method, the nonlinear signal
modulation technique that provides the explicit control
over the signal extent. Our approach ensures that the
time-domain profile corresponding to the b-modulated data
has a limited duration, including the cases when the
bound states (discrete solitonic eigenvalues) are present.
In particular, in contrast to the previous approaches, we
show that it is possible to include the discrete eigenvalues
with the specially chosen parameters into the b-modulation
concept while keeping the signal localization property
exactly.