Quantized vortex dynamics of the nonlinear wave equation on the torus

We rigorously derive the reduced dynamical law for quantized vortex dynamics of the nonlinear wave equation on the torus when the core size of vortex $ \varepsilon\to 0 $. It is proved that the reduced dynamical law is a system consisting of second-order nonlinear ordinary differential equations driven by the renormalized energy on the torus, and the initial data of the reduced dynamical law is determined by the positions of vortices and the limit momentum of the solution of the nonlinear wave equation. We will also investigate the effect of the limit momentum on the vortex dynamics via numerical simulation.