Higher order asymptotic analysis of the Klein-Gordon equation in the non-relativistic limit regime

In this paper, we study the asymptotic behavior of nonlinear Klein-Gordon equations in the non-relativistic limit regime. By employing the techniques in geometric optics, we show that the Klein-Gordon equation can be approximated by nonlinear Schrdinger equations. In particular, we show error estimates which are of the same order as the initial error. Our result gives a mathematical verification for some numerical results obtained in [SIAM J. Numer. Anal. 52 (2014), 2488-2511] and [Numer. Math. 120 (2012), 189-229], and offers a rigorous justification for a technical assumption in the numerical studies [SIAM J. Numer. Anal. 52 (2014), 2488-2511].