Classification of dark solitons via topological vector potentials

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

Figure 1

(a) Topological vector potential $\mathbf{A}$ for a scalar dark soliton. (b) Corresponding magnetic field $\mathbf{B}$, and (c) the phase jump induced by the magnetic field. The soliton speed is set to $v=0.50$. The pointlike magnetic fields are located at $(0,\pm 0.61)$ with a period of $1.15\pi$ along the imaginary axis.

Figure 2

(a) Two-step structure of the phase jump and its corresponding magnetic fields in the complex plane for the symmetric DBBS1 soliton. The soliton speed is $v=0.50$. The pointlike magnetic fields are located at $(\pm 0.61,\pm 0.35)$ with a period of $\pi$ along the imaginary axis. The subgraphs show the phase variations induced by the magnetic fields on each line. (b) Triple-step structure of the phase jump and its corresponding magnetic fields in the complex plane for the symmetric DBBBS soliton. The soliton speed is $v=0.42$. The pointlike magnetic fields are located at $(\pm 0.99,\pm 0.27)$, $(0,\pm 0.18)$ with a period of $\pi$ along the imaginary axis. The subgraphs show the phase variations induced by the magnetic fields on each line.

Figure 3

(a) Topological vector potential $\mathbf{A}$ for DBBS2 for which the dark soliton admits an asymmetric double valley. (b) Phase jumps of the dark soliton and the corresponding magnetic field distribution. The singularity is located at $(-0.26,\pm 0.33)$, $(-0.04,\pm 2.21)$, $(0.30,\pm 2.49)$, $(1.54,\pm 2.59)$, and $(1.67,\pm 0.41)$ with a period of $2\pi$ along the imaginary axis. The soliton speed is $v=0.50$. The subgraphs show the phase variations induced by the singular magnetic fields on the five separate lines.