{"title":"修正一维Schrödinger方程的全局解","authors":"Ting Zhang","doi":"10.4208/cmr.2021-0015","DOIUrl":null,"url":null,"abstract":"In this paper, we consider the modified one-dimensional Schrödinger equation: ( Dt−F(D) ) u=λ|u|u, where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size ε≪1. We prove that the solution is global-intime, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when t→+∞. AMS subject classifications: Primary: 35Q55; Secondary: 35A01, 35B40, 35S50","PeriodicalId":66427,"journal":{"name":"数学研究通讯","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Global Solutions of Modified One-Dimensional Schrödinger Equation\",\"authors\":\"Ting Zhang\",\"doi\":\"10.4208/cmr.2021-0015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we consider the modified one-dimensional Schrödinger equation: ( Dt−F(D) ) u=λ|u|u, where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size ε≪1. We prove that the solution is global-intime, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when t→+∞. AMS subject classifications: Primary: 35Q55; Secondary: 35A01, 35B40, 35S50\",\"PeriodicalId\":66427,\"journal\":{\"name\":\"数学研究通讯\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"数学研究通讯\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4208/cmr.2021-0015\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"数学研究通讯","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4208/cmr.2021-0015","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Global Solutions of Modified One-Dimensional Schrödinger Equation
In this paper, we consider the modified one-dimensional Schrödinger equation: ( Dt−F(D) ) u=λ|u|u, where F(ξ) is a second order constant coefficients classical elliptic symbol, and with smooth initial datum of size ε≪1. We prove that the solution is global-intime, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we get a one term asymptotic expansion for u when t→+∞. AMS subject classifications: Primary: 35Q55; Secondary: 35A01, 35B40, 35S50