  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
classic Nyquist-based schemes. This is especially due to the observation that
soliton pulses generally exhibit a large time-bandwidth product. We consider
the phase modulation of spectral amplitudes of higher order solitons, taking
into account their varying spectral and temporal behavior when propagating
along the fiber. For second and third order solitons, we numerically optimize
the pulse shapes to minimize the time-bandwidth product. We study the behavior
of multi-soliton pulse duration and bandwidth and generally observe two corner
cases where we approximate them analytically. We use these results to give an
estimate on the minimal achievable time-bandwidth product per eigenvalue.
