Points or loops where phase winds by 2n2\pi n2n (with nZn \in \mathbb{Z}nZ): indicating quantized vortices
Phase singularities where amplitude vanishes A(x,t)0A(x,t) \to 0A(x,t)0: locations of topological charge
Persistent vortex-antivortex pairs, forming as a consequence of high-energy blink excitation
These structures resemble topological excitations in superfluids and Bose-Einstein condensates, where phase continuity constraints enforce quantization.
2. Soliton and Bubble-Like Configurations
Nonlinear coupling I2I\lambda |I|^2 II2I supports the formation of soliton-like envelopes---spatially localized, temporally coherent structures with preserved shape due to balance between nonlinearity and dispersion:
1D simulations: bright and dark solitons, depending on pulse sign
2D/3D simulations: bubble-like shells or filamentary strings, some enclosing nontrivial topologies
These solitonic textures act as non-perturbative field configurations, potentially encoding information memory or field parity. They survive long after the initiating blink has decayed.
3. Topological Charge and Stability
To characterize these excitations, we define a topological charge density in 2D:
q(x,y,t)=12(xyyx)q(x,y,t) = \frac{1}{2\pi} \left( \partial_x \phi \cdot \partial_y \phi - \partial_y \phi \cdot \partial_x \phi \right)q(x,y,t)=21(xyyx)
And integrated topological charge:
Q=q(x,y,t)dxdyQ = \int q(x,y,t)\, dx\, dyQ=q(x,y,t)dxdy
Results:
In regions with isolated vortices: Q1Q \approx \pm 1Q1
For vortex-antivortex pairs: Q0Q \to 0Q0, but internal tension preserved
In turbulent excitation: complex patterns of transiently bound topological clusters
These charges are conserved under continuous deformation, showcasing topological robustness of the emergent features.
