Quantized vortex dynamics of the nonlinear Schrödinger equation on torus with non-vanishing momentum

Abstract

We derive rigorously the reduced dynamical law for quantized vortex dynamics of the nonlinear Schrödinger equation on the torus with non-vanishing momentum when the vortex core size ɛ→0. The reduced dynamical law is governed by a Hamiltonian flow driven by a renormalized energy. A key ingredient is to construct a new canonical harmonic map to include the effect from the non-vanishing momentum into the dynamics. Finally, some properties of the reduced dynamical law are discussed.