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We study the the nonlinear Klein-Gordon (NLKG) equation on a manifold $M$ inthe nonrelativistic limit, namely as the speed of light $c$ tends to infinity.In particular, we consider an order-$r$ normalized approximation of NLKG (whichcorresponds to the NLS at order $r=1$), and prove that when $M=\mathbb{R}^d$,$d \geq 2$, small radiation solutions of the order-$r$ normalized equationapproximate solutions of the nonlinear NLKG up to times of order$\mathcal{O}(c^{2(r-1)})$.

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