  Based on the $\phi$-mapping theory, we obtain an exact Bogomol'nyi self-dual
equation with a topological term, which is ignored in traditional self-dual
equation, in the fractional quantum Hall system. It is revealed that there
exist self-dual vortices in the system. We investigate the inner topological
structure of the self-dual vortices and show that the topological charges of
the vortices are quantized by Hopf indices and Brouwer degrees. Furthermore, we
study the branch processes in detail. The vortices are found generating or
annihilating at the limit points and encountering, splitting or merging at the
bifurcation points of the vector field $\vec\phi$.
