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

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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 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.

Original languageEnglish
Pages (from-to)584-589
Number of pages6
JournalJournal of Mathematical Analysis and Applications
Volume365
Issue number2
DOIs
Publication statusPublished - 2010 May 15

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All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

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