Convergence of the Klein–Gordon equation to the wave map equation with magnetic field

This paper is devoted to the proof of the convergence from the modulated cubic nonlinear defocusing Klein–Gordon equation with magnetic field to the wave map equation. More precisely, we discuss the nonrelativistic-semiclassical limit of the modulated cubic nonlinear Klein–Gordon equation with magnetic field where the Planck's constant ℏ=ε and the speed of light c are related by c=ε−α for some α⩾1. When α=1 the limit wave function satisfies the wave map with one extra term coming from the magnetic field. However, α>1, the effect of the magnetic filed disappears and the limit is the typical wave map equation only.