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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 \(\mathbb {R}^3\).

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