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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics