  A fully analytical theory of a traveling soliton in a one-dimensional
fermionic superfluid is developed within the framework of time-dependent
self-consistent Bogoliubov-de Gennes equations, which are solved exactly in the
Andreev approximation. The soliton manifests itself in a kink-like profile of
the superconducting order parameter and hosts a pair of Andreev bound states in
its core. They adjust to soliton's motion and play an important role in its
stabilization. A phase jump across the soliton and its energy decrease with
soliton's velocity and vanish at the critical velocity, corresponding to the
Landau criterion, where the soliton starts emitting quasiparticles and becomes
unstable. The "inertial" and "gravitational" masses of the soliton are
calculated and the former is shown to be orders of magnitude larger than the
latter. This results in a slow motion of the soliton in a harmonic trap,
reminiscent to the observed behavior of a soliton-like texture in related
experiments in cold fermion gases [T. Yefsah et al., Nature 499, 426, (2013)].
Furthermore, we calculate the full non-linear dispersion relation of the
soliton and solve the classical equations of motion in a trap. The strong
non-linearity at high velocities gives rise to anharmonic oscillatory motion of
the soliton. A careful analysis of this anharmonicity may provide a means to
experimentally measure the non-linear soliton spectrum in superfluids.
