On the NLS dynamics for infinite energy vortex configurations on the plane (Q951995)

The aim of this paper is to study the dynamics of vortices for the nonlinear Schrödinger equation on the plane, when the total degree at infinity is nonzero. The equation which the authors are interested in is written on \(\mathbb{R}^2\times\mathbb{R}\) as \[ i\partial_tu_\varepsilon+\Delta u_\varepsilon=\frac{1} {\varepsilon^2}\left(| u_\varepsilon|^2-1\right)\cdot u_\varepsilon,\tag{1} \] where \(0<\varepsilon<1\) denotes a small parameter. The authors derive the asymptotical dynamical law.