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

Abstract

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 h = epsilon and the speed of light c are related by c = epsilon(-alpha) for some alpha >= 1. When alpha = 1 the limit wave function satisfies the wave map with one extra term coming from the magnetic field. However, alpha > 1, the effect of the magnetic filed disappears and the limit is the typical wave map equation only. (C) 2009 Elsevier Inc. All rights reserved.