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

## TL;DR

This paper investigates the behavior of the nonlinear Klein-Gordon equation in the nonrelativistic limit, constructing higher-order approximations and proving their accuracy in various geometric settings, including Euclidean space.

## Contribution

It introduces higher-order normalized approximations of NLKG and proves their local uniform approximation to NLKG solutions as the speed of light tends to infinity.

## Key findings

- Higher-order normalized approximations effectively approximate NLKG solutions.
- Solutions of the linearized normalized equations approximate linear Klein-Gordon solutions over extended times.
- The approximation accuracy depends on the order of the normalized equation and the geometry of the manifold.

## Abstract

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 $M=\mathbb{R}^d$, $d \geq 3$, 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 $r>1$.