Dynamics of the nonlinear Klein–Gordon equation in the nonrelativistic limit

Abstract

We study the nonlinear Klein–Gordon (NLKG) equation on a manifold M in the nonrelativistic limit, namely as the speed of light c tends to infinity. In particular, we consider a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $$r=1$$ ) and prove that when M is a smooth compact manifold or $$\mathbb {R}^d$$ , the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $$M=\mathbb {R}^d$$ , $$d \ge 2$$ , we also prove that for $$r \ge 2$$ small radiation solutions of the order-r normalized equation approximate solutions of the nonlinear NLKG up to times of order $$\mathscr {O}(c^{2(r-1)})$$ . We also prove a global existence result uniform with respect to c for the NLKG equation on $$\mathbb {R}^3$$ with cubic nonlinearity for small initial data and Strichartz estimates for the Klein–Gordon equation with potential on $$\mathbb {R}^3$$ .