Revista Matemática Iberoamericana

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Volume 24, Issue 2, 2008, pp. 671–702
DOI: 10.4171/RMI/552

Published online: 2008-08-31

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

Fabrice Bethuel[1], Robert L. Jerrard[2] and Didier Smets[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) University of Toronto, Canada
(3) UPMC, Université Paris 06, France

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.

Keywords: Vortex dynamics, NLS equation, superfluids

Bethuel Fabrice, Jerrard Robert, Smets Didier: On the NLS dynamics for infinite energy vortex configurations on the plane. Rev. Mat. Iberoam. 24 (2008), 671-702. doi: 10.4171/RMI/552