Convergence of the Klein-Gordon equation to the wave map equation with magnetic field |
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Authors: | Kung-Chien Wu |
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Affiliation: | Department of Applied Mathematics, Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu 30010, Taiwan |
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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 ?=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. |
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Keywords: | Klein-Gordon equation Magnetic field Wave map equation Nonrelativistic-semiclassical limit |
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