On the NLS dynamics for infinite energy vortex configurations on the plane

Abstract

We derive the asymptotical dynamical law for Ginzburg-Landau vortices in the plane under the Schrödinger dynamics, as the Ginz\-burg-Landau parameter goes to zero. The limiting law is the well-known point-vortex system. This result extends to the whole plane previous results of [Colliander, J.E. and Jerrard, R.L.: Vortex dynamics for the Ginzburg-Landau-Schrödinger equation. Internat. Math. Res. Notices 1998, no. 7, 333-358; Lin, F.-H. and Xin, J.\,X.: On the incompressible fluid limit and the vortex motion law of the nonlinear Schr\"{o}dinger equation. Comm. Math. Phys. 200 (1999), 249-274] established for bounded domains, and holds for arbitrary degree at infinity. When this degree is non-zero, the total Ginzburg-Landau energy is infinite.