  Multi-soliton pulses are potential candidates for fiber optical transmission
where the information is modulated and recovered in the so-called nonlinear
Fourier domain. While this is an elegant technique to account for the channel
nonlinearity, the obtained spectral efficiency, so far, is not competitive with
the classic Nyquist-based schemes. In this paper, we study the evolution of the
time-bandwidth product of multi-solitons as they propagate along the optical
fiber. For second and third order soliton pulses, we numerically optimize the
pulse shapes to achieve the smallest time-bandwidth product when the phase of
the spectral amplitudes is used for modulation. Moreover, we analytically
estimate the pulse-duration and bandwidth of multi-solitons in some practically
important cases. Those estimations enable us to approximate the time-bandwidth
product for higher order solitons.
