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 Ginzburg–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 Schrödinger 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.