In the study of superfluid systems, vortices are often referred to as "topological excitations", because the winding of the phase of the superfluid order parameter around a vortex is a topological property which must be conserved. On the other hand, dark solitons are also known to have a characteristic phase profile: they form a boundary between two regions with a different global phase, creating either a discrete or continuous phase jump at the position of the soliton. Since in some general texts about solitons I have come across the notion of a "topological soliton", my question is the following:

Can a dark soliton in a superfluid also be considered a topological excitation? If so, is the phase jump across the soliton plane the topological invariant and why exactly must it be conserved?


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