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In this note, a brief introduction to the physical and mathematical background of the two-component Ginzburg-Landau theory is given. From this theory we derive a boundary value problem whose solution can be obtained in part by solving a minimization problem using the technique of variational method, except that two of the eight boundary conditions cannot be satisfied. To overcome the difficulty of recovering the full set of boundary conditions, we employ a variety of methods, including the uniform estimation method and the bounded monotonic theorem, which may be applied to other complicated vortex problems in gauge field theories. The twisted vortex solutions are obtained as energy-minimizing cylindrically symmetric field configurations. We also give the sharp asymptotic estimates for the twisted vortex solutions at the origin and infinity.
Cao et al. (Mon,) studied this question.