Classification of dark solitons via topological vector potentials

Dark solitons are some of the most interesting nonlinear excitations and considered to be one-dimensional topological analogs vortices. However, in contrast their two-dimensional vortex counterparts, characteristics a dark soliton far from fully understood because charge defined according phase jump cannot reflect its essential property. Here, similar complex extension used exploration partition-function zeros depict thermodynamic states, we extend coordinate space explore density solitons. Surprisingly find that these constitute pointlike magnetic fields, each which has quantized flux elementary $\ensuremath{\pi}$. The corresponding vector potential fields demonstrate topology Wess-Zumino term can Then classify Euler characteristic manifold fields. Our study not only reveals features but also applied identify new with high complexity.