Dynamics of the nonlinear Klein-Gordon equation in the nonrelativistic limit, I

The nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ in the nonrelativistic limit, namely as the speed of light $c$ tends to infinity, is considered. In particular, a higher-order normalized approximation of NLKG (which corresponds to the NLS at order $r=1$) is constructed, and when $M$ is a smooth compact manifold or ${\mathbb{R}}^d$ it is proved that the solution of the approximating equation approximates the solution of the NLKG locally uniformly in time. When $M={\mathbb{R}}^d$, $d\ge 3$, it is proved that solutions of the linearized order $r$ normalized equation approximate solutions of linear Klein-Gordon equation up to times of order $\mathcal{O}(c^{2(r-1)})$ for any $r>1$.