Quantized Vortex Dynamics of the Coupled Nonlinear Schr\"odinger Equation

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the coupled nonlinear Schr\"odinger equation without Josephson junction (CNLS) when the core size of vortex $\varepsilon\to 0$. It is proved that when $\varepsilon\to 0$, the vortex motion of one component won't affect the vortex motion on the other component. Moreover, the motion of vortices of each component follows the vortex motion law for the nonlinear Schr\"odinger.