The topological quantization and bifurcation of the topological linear defects

TL;DR

This paper investigates the topological structure, quantization, and bifurcation phenomena of linear defects using $$-mapping and topological current theory, revealing conditions for defect origin and bifurcation points.

Contribution

It introduces a detailed analysis of the topological quantization and bifurcation of linear defects, especially near limit and bifurcation points, using advanced mathematical methods.

Findings

Existence of topological quantization of linear defects.

Identification of branch processes at critical Jacobian conditions.

Detailed description of defect origin and bifurcation near special points.

Abstract

In the light of $\phi$-mapping method and topological current theory, the topological structure and the topological quantization of topological linear defects are obtained under the condition that the Jacobian $J(\phi/v) \neq 0$. When $J(\phi/v) = 0$, it is shown that there exist the crucial case of branch process. Based on the implicit function theorem and the Taylor expansion, the origin and bifurcation of the linear defects are detailed in the neighborhoods of the limit points and bifurcation points of $\phi$-mapping, respectively.