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Chin. Phys. B, 2012, Vol. 21(9): 090304    DOI: 10.1088/1674-1056/21/9/090304
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Topological aspect of vortex lines in two-dimensional Gross–Pitaevskii theory

Zhao Li (赵力)a, Yang Jie (杨捷)a, Xie Qun-Ying (谢群英)a b, Tian Miao (田苗)c
a Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China;
b School of Information Science and Engineering, Lanzhou University, Lanzhou 730000, China;
c School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou 730000, China
Abstract  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 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.
Keywords:  Gross-Pitaevskii equation      Bose-Einstein condensate      vortex line      bifurcation theory  
Received:  25 August 2011      Revised:  14 April 2012      Accepted manuscript online: 
PACS:  03.75.Lm (Tunneling, Josephson effect, Bose-Einstein condensates in periodic potentials, solitons, vortices, and topological excitations)  
  47.32.C- (Vortex dynamics)  
  02.40.Pc (General topology)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 10905026 and 10905027) and the Program of Science and Technology Development of Lanzhou, China (Grant No. 2010-1-129).
Corresponding Authors:  Zhao Li     E-mail:  lizhao@lzu.edu.cn

Cite this article: 

Zhao Li (赵力), Yang Jie (杨捷), Xie Qun-Ying (谢群英), Tian Miao (田苗) Topological aspect of vortex lines in two-dimensional Gross–Pitaevskii theory 2012 Chin. Phys. B 21 090304

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