We study dynamics of vortices in solutions of the Gross-Pitaevskii equation i∂tu = Δu + ϵ−2u(1 − |u|2) on ℝ2with nonzero degree at infinity. We prove that vortices move according to the classical Kirchhoff-Onsager ODE for a small but finite coupling parameter ϵ. 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 .
- Euler equations
- Point vortex method