TY - JOUR

T1 - Singular limits of the Klein-Gordon equation

AU - Lin, Chi Kun

AU - Wu, Kung Chien

N1 - Funding Information:
This work is partially supported by National Science Council of Taiwan under the grant NSC98-2115-M-009-004-MY3.

PY - 2010

Y1 - 2010

N2 - 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, H 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, H 0, c = H-α ∞ 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.

AB - 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, H 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, H 0, c = H-α ∞ 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.

UR - http://www.scopus.com/inward/record.url?scp=77954175840&partnerID=8YFLogxK

U2 - 10.1007/s00205-010-0324-8

DO - 10.1007/s00205-010-0324-8

M3 - Article

AN - SCOPUS:77954175840

SN - 0003-9527

VL - 197

SP - 689

EP - 711

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 2

ER -