Quantized vortex dynamics of the complex Ginzburg-Landau equation on torus

We derive rigorously the reduced dynamical laws for quantized vortex dynamics of the complex Ginzburg-Landau equation on torus when the core size of vortex $\epsilon \to 0$. The reduced dynamical laws of the complex Ginzburg-Landau equation are governed by a mixed flow of gradient flow and Hamiltonian flow which are both driven by a renormalized energy on torus. Finally, some first integrals and analytic solutions of the reduced dynamical laws are discussed.