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 Schrödinger 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].
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