Singular limits of the Klein-Gordon equation

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.