The New Applications Of φ-Mapping Topological Current Theory In The Physics

In this thesis, using (?)-mapping topological current theory and U(1) gauge potential theory, we study the Bifurcation Theory of Magnetic Monopoles in a Charged Two-Condensate Bose-Einstein System, the branch processes of p-branes and knot Solitons in AFZ model.First, in charged two-component Bose-Einstein system, we obtained the dynamic form of magnetic monopole and quantized the magnetic charge at the topological level in units of(?). The topological quantum numbers are determined by the Hopf indices and Brouwer degrees, which are topological numbers. The evolution of magnetic monopoles is studied from the topological properties of a three-dimensional vector field(?). We find that there exist crucial cases of branch processes in the evolution of the magnetic monopoles when D(?), i.e.,(?) is indefinite. This means that the magnetic monopoles generate or annihilate at the limit points and encounter, split, or merge at the bifurcation points of the three-dimensional vector field Z, which shows that the magnetic monopoles system is unstable at these branch points. We show the result that the velocity of magnetic monopole is infinite when they are annihilating or generating. Furthermore, we must point out that there exist two restrictions of the evolution of magnetic monopoles. One restriction is the conservation of the topological charge of the magnetic monopoles during the branch process, the other is that the number of different directions of the world lines of magnetic monopoles is at most 4 at the bifurcation points.Second, We display that the p-branes are generated from the zero points of the order parameter field (?), and their topological charges are quantized in term of the Brouwer degrees and Hopf indices of 0-mapping under the condition that the zero points of field (?) are regular points. While at the critical points of the order parameter field (?), i.e., the limit points and bifurcation points, there exist branch processes, the topological current of defect bifurcates and the p-branes split or merge at such point, which mean that p-branes are unstable at these points.Last, In this paper, we mainly focus on the knot solitons in AFZ model. Using U(1) gauge potential theory and Duan's topological current theory, we study the topological properties of knot solitons in AFZ model in detail and naturally obtain the inner structure of the nontrivial Hopf invariant. Furthermore, by comparing with others, this paper considers not only linking numbers but also self-linking numbers in detail.