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Singular Limits of the Klein–Gordon Equation

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Abstract

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, \({\hbar\to 0}\), we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, \({\hbar\to 0, c=\hbar^{-\alpha}\to \infty}\) for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.

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Authors and Affiliations

  1. Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu, 30010, Taiwan

    Chi-Kun Lin & Kung-Chien Wu

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  1. Chi-Kun Lin
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  2. Kung-Chien Wu
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Correspondence to Chi-Kun Lin.

Additional information

Communicated by T.-P. Liu

Dedicated to Professor Yuh-Jia Lee on his sixtieth birthday.

This work is partially supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3.

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Lin, CK., Wu, KC. Singular Limits of the Klein–Gordon Equation. Arch Rational Mech Anal 197, 689–711 (2010). https://doi.org/10.1007/s00205-010-0324-8

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