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

Article type: Research Article

Authors: Lu, Yong^{a; *} | Zhang, Zhifei^b

Affiliations: [a] Chern Institute of Mathematics & LPMC, Nankai University, Tianjin 300071, China. E-mail: [email protected] | [b] School of Mathematical Science, Peking University, Beijing 100871, China. E-mail: [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: 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].

Keywords: Klein–Gordon equation, non-relativistic limit

DOI: 10.3233/ASY-171414

Journal: Asymptotic Analysis, vol. 102, no. 3-4, pp. 157-175, 2017