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

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the complex GinzburgLandau equation on the torus when the core size of vortex epsilon -> 0. The reduced dynamical law of the complex Ginzburg-Landau equation is governed by a mixed flow of gradient flow and Hamiltonian flow which are both driven by a renormalized energy on the torus. Finally, some first integrals and analytic solutions of the reduced dynamical law are discussed. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.