Hydrodynamic limit of the Gross-Pitaevskii equation

Abstract

We study dynamics of vortices in solutions of the Gross-Pitaevskii equation $i{\partial }_{t}u=\Delta u+{\epsilon }^{-2}u(1-|u{|}^2)$ on ${\mathbb{R}}^2$ with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter $\epsilon$. By carefully tracking errors we allow for asymptotically large numbers of vortices, and this lets us connect the Gross-Pitaevskii equation on the plane to two dimensional incompressible Euler equations through the work of Schochet [21].

References in corpus (1)

• Vortex Dynamics for the Ginzburg-Landau-Schrödinger Equation