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Topological aspect of vortex lines in two-dimensional Gross—Pitaevskii theory

Using the φ-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross—Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross—Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose—Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the δ function indicates that the vortices can only be generated from the zero points of Φ and are quantized in terms of the Hopf indices and Brouwer degrees. The φ-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross—Pitaevskii theory.