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 -