  Dark soliton is usually seen as one of the simplest topological solitons, due
to the phase jump across its density dip. We investigate the phase jump
properties of dark solitons in a single mode optical fiber with the third-order
dispersion and delayed nonlinear response, based on exact analytical solutions
of Hirota equation. Our analysis indicates that a single-valley dark soliton
(SVDS) can admit two distinct phase jumps at the same velocity, in sharp
contrast to the dark soliton with only the second-order dispersion and
self-phase modulation, which admits a one-to-one match between the velocity and
phase jump. We further uncover the different topological vector potentials
underlying the distinct phase jumps. The relations between phase jump and
velocity of the SVDS can explain the generation of the previously reported
double-valley dark soliton (DVDS). The detailed analysis on the two phase jumps
characters of the DVDS with one identical velocity enables us to obtain
U-shaped type or double-step type phase distribution. We further explore
collision properties of the DVDSs by analyzing their topological phase, which
can be considered as the generalization of topological phase (Phys. Rev. E 103,
L040204). Strikingly, the inelastic collision can lead to the conversion
between the two types of phase distributions for DVDS. The results reveal that
inelastic or elastic collision can be judged by analyzing the magnetic monopole
fields.
