  Yrast states are the lowest energy states at given non-zero momentum and
provide a natural extension of the concept of dark solitons to
strongly-interacting one-dimensional quantum gases. Here we study the yrast
states of the balanced spin-$\frac{1}{2}$ Fermi gas with attractive
delta-function interactions in one dimension with the exactly solvable
Yang-Gaudin model. The corresponding Bethe-ansatz equations are solved for
finite particle number and in the thermodynamic limit. Properties corresponding
to the soliton-like nature of the yrast excitations are calculated including
the missing particle number, phase step, and inertial and physical masses. The
inertial to physical mass ratio, which is related to the frequency of
oscillations in a trapped gas, is found to be unity in the limits of strong and
weak attraction and falls to $\approx 0.78$ in the crossover regime. This
result is contrasted by one-dimensional mean field theory, which predicts a
divergent mass ratio in the weakly attractive limit. By means of an exact
mapping our results also predict the existence and properties of
dark-soliton-like excitations in the super Tonks-Girardeau gas. The prospects
for experimental observations are briefly discussed.
