  In this paper, the two-component Ginzburg-Landau (TCGL) model, which is an
important research object in mathematics and physics of two scalar fields with
$U(1)\times U(1)$ symmetry, is studied in detail by the shooting method and the
fixed point theorem, which contains two cases: single vacuum expectation value
(1VEV) and two vacuum expectation values (2VEV). This represents two different
physical states: superconductivity-normal state and superconductivity
superconductivity state, and we prove the existence, uniqueness, asymptotic
properties and quantization of the solutions to these two boundary value
problems.
