Topological defect system in O(n) symmetric time-dependent Ginzburg-Landau model

We present a new generalized topological current in terms of the order parameter field $\vec \phi$ to describe the topological defect system in O(n) symmetric time-dependent Ginzburg-Landau model. With the aid of the $% \phi$-mapping method, the structure of the topological defects and the topological quantization of their topological charges in TDGL model are obtained under the condition that the Jacobian $% J(\frac \phi v)\neq 0$. We show that the topological defects are generated from the zero points of the order parameter field $\vec \phi$, and the topological charges of these topological defects are topological quantized in terms of the Hopf indices and Brouwer degrees of $\phi$-mapping under the condition. When $J(\frac \phi v)=0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, we detail the bifurcation of generalized topological current and find different directions of the bifurcation. The topological defects in TDGL model are found splitting or merging at the degenerate point of field function $\vec \phi $ but the total charge of the topological defects is still unchanged.