### 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 language | English |
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Pages (from-to) | 584-589 |

Number of pages | 6 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 365 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2010 May 15 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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**Convergence of the Klein-Gordon equation to the wave map equation with magnetic field.** / Wu, Kung-Chien.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Convergence of the Klein-Gordon equation to the wave map equation with magnetic field

AU - Wu, Kung-Chien

PY - 2010/5/15

Y1 - 2010/5/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=74849083670&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=74849083670&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.11.022

DO - 10.1016/j.jmaa.2009.11.022

M3 - Article

VL - 365

SP - 584

EP - 589

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -