Classification of Dark Solitons via Topological Vector Potentials

Dark soliton is one of most interesting nonlinear excitations in physical systems, manifesting a spatially localized density "dip" on a uniform background accompanied with a phase jump across the dip. However, the topological properties of the dark solitons are far from fully understood. Our investigation for the first time uncover a vector potential underlying the nonlinear excitation whose line integral gives the striking phase jump. More importantly, we find that the vector potential field has a topological configuration in analogous to the Wess-Zumino term in a Lagrangian representation. It can induce some point-like magnetic fields scattered periodically on a complex plane, each of them has a quantized magnetic flux of elementary $\pi$. We then calculate the Euler characteristic of the topological manifold of the vector potential field and classify all known dark solitions according to the index.