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

TL;DR

This paper investigates the behavior of the nonlinear Klein-Gordon equation on manifolds as the speed of light approaches infinity, establishing approximation results for normalized solutions over long times in the nonrelativistic limit.

Contribution

It introduces an order-$r$ normalized approximation of NLKG and proves its solutions closely approximate NLKG solutions over extended times in the nonrelativistic limit.

Findings

Normalized solutions approximate NLKG solutions up to long times

Approximation accuracy improves with higher order normalization

Results apply to manifolds with dimension $d \\geq 2$

Abstract

We study the 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 an order-$r$ normalized approximation of NLKG (which corresponds 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 equation approximate solutions of the nonlinear NLKG up to times of order $\mathcal{O}(c^{2(r-1)})$.