{"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}
引用次数: 3
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