Phase characters of optical dark solitons with third-order dispersion and delayed nonlinear response

Dark soliton is usually seen as one of the simplest topological solitons, due to phase shift across its intensity dip. We investigate phase characters of single-valley dark soliton (SVDS) and double-valley dark soliton (DVDS) in a single-mode optical fiber with third-order dispersion and delayed nonlinear response. Notably, two different phase shifts can produce an SVDS with the same velocity under some conditions, which is not admitted for a dark soliton with only the second-order dispersion and self-phase modulation, whose phase shift and velocity is a one-to-one match. This phase property of SVDS can be used to explain the generation of previously reported DVDS in Hirota equation and make DVDSs show two types of phase profiles. Moreover, the different topological vector potentials underlying the distinct phase profiles have been uncovered. We further explore the collision properties of the DVDSs by analyzing their topological phases. Strikingly, the inelastic collision can lead to the conversion between the two types of phase profiles for DVDS. The results reveal that inelastic or elastic collision can be judged by analyzing virtual topological magnetic fields.