{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,5]],"date-time":"2026-01-05T20:39:46Z","timestamp":1767645586550,"version":"3.48.0"},"reference-count":21,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2017,4,14]],"date-time":"2017-04-14T00:00:00Z","timestamp":1492128000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2017,4,14]]},"abstract":"<jats:p>\n                    In this paper, we study the asymptotic behavior of nonlinear Klein\u2013Gordon equations in the non-relativistic limit regime. By employing the techniques in geometric optics, we show that the Klein\u2013Gordon equation can be approximated by nonlinear Schr\u00f6dinger 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 [\n                    <jats:italic toggle=\"yes\">SIAM J. Numer. Anal.<\/jats:italic>\n                    <jats:bold>52<\/jats:bold>\n                    (\n                    <jats:xref ref-type=\"bibr\">2014<\/jats:xref>\n                    ), 2488\u20132511] and [\n                    <jats:italic toggle=\"yes\">Numer. Math.<\/jats:italic>\n                    <jats:bold>120<\/jats:bold>\n                    (\n                    <jats:xref ref-type=\"bibr\">2012<\/jats:xref>\n                    ), 189\u2013229], and offers a rigorous justification for a technical assumption in the numerical studies [\n                    <jats:italic toggle=\"yes\">SIAM J. Numer. Anal.<\/jats:italic>\n                    <jats:bold>52<\/jats:bold>\n                    (\n                    <jats:xref ref-type=\"bibr\">2014<\/jats:xref>\n                    ), 2488\u20132511].\n                  <\/jats:p>","DOI":"10.3233\/asy-171414","type":"journal-article","created":{"date-parts":[[2017,4,14]],"date-time":"2017-04-14T11:37:59Z","timestamp":1492169879000},"page":"157-175","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Higher order asymptotic analysis of the Klein\u2013Gordon equation in the non-relativistic limit regime"],"prefix":"10.1177","volume":"102","author":[{"given":"Yong","family":"Lu","sequence":"first","affiliation":[{"name":"Nankai University","place":["China"]}]},{"given":"Zhifei","family":"Zhang","sequence":"additional","affiliation":[{"name":"Peking University","place":["China"]}]}],"member":"179","published-online":{"date-parts":[[2017,4,14]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1137\/130950665"},{"key":"e_1_3_2_3_2","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-011-0411-2"},{"key":"e_1_3_2_4_2","doi-asserted-by":"publisher","DOI":"10.3934\/dcds.2004.11.83"},{"key":"e_1_3_2_5_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01168155"},{"key":"e_1_3_2_6_2","doi-asserted-by":"publisher","DOI":"10.1016\/s0294-1449(16)30329-8"},{"key":"e_1_3_2_7_2","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.2000.3794"},{"key":"e_1_3_2_8_2","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-1998-318"},{"key":"e_1_3_2_9_2","doi-asserted-by":"publisher","DOI":"10.3233\/ASY-2012-1140"},{"key":"e_1_3_2_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2015.10.001"},{"key":"e_1_3_2_11_2","unstructured":"Y.\u00a0Lu and B.\u00a0Texier A stability criterion for high-frequency oscillations M\u00e9m. 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