Classification of dark solitons via topological vector potentials

Area of Science:

• Nonlinear physics
• Topological phenomena
• Quantum field theory

Background:

• Dark solitons are 1D topological analogs of vortices, but their topological characteristics remain poorly understood.
• The conventional phase jump definition of topological charge fails to capture essential properties of dark solitons.

Purpose of the Study:

• To develop a novel method for characterizing the topology of dark solitons.
• To reveal the underlying topological features and enable classification of dark solitons.

Main Methods:

• Extended the complex coordinate space to analyze the density zeros of dark solitons.
• Investigated the resulting magnetic fields and vector potential fields.
• Classified dark solitons based on the Euler characteristic of the vector potential field's topological manifold.

Main Results:

• Discovered that dark soliton density zeros correspond to pointlike magnetic fields, each with a quantized magnetic flux of π.
• Demonstrated that the derived vector potential fields exhibit the topology of the Wess-Zumino term.
• Successfully classified dark solitons using the Euler characteristic of these topological manifolds.

Conclusions:

• The study reveals the intrinsic topological features of dark solitons through their density zeros.
• The developed framework provides a new approach to explore and identify dark solitons with complex topological properties.