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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 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 [1, 2], and offers a rigorous justification for a technical assumption in the numerical studies [1].