{"title":"Asymptotic behavior in time of solution for the cubic nonlinear Schrödinger equation on the tadpole graph","authors":"Jun-ichi Segata","doi":"10.1016/j.jde.2024.11.006","DOIUrl":null,"url":null,"abstract":"<div><div>The purpose of this paper is to study large time behavior of solution to the cubic nonlinear Schrödinger equation on the tadpole graph which is a ring attached to a semi-infinite line subject to the Kirchhoff conditions at the junction. Note that the cubic nonlinearity belongs borderline between short and long range scatterings on the whole line. We show that if the initial data has some symmetry on the graph which excludes the standing wave solutions, then the asymptotic behavior of solution to this equation is given by the solution to linear equation with logarithmic phase correction by the nonlinear effect.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1977-1999"},"PeriodicalIF":2.4000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S002203962400723X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The purpose of this paper is to study large time behavior of solution to the cubic nonlinear Schrödinger equation on the tadpole graph which is a ring attached to a semi-infinite line subject to the Kirchhoff conditions at the junction. Note that the cubic nonlinearity belongs borderline between short and long range scatterings on the whole line. We show that if the initial data has some symmetry on the graph which excludes the standing wave solutions, then the asymptotic behavior of solution to this equation is given by the solution to linear equation with logarithmic phase correction by the nonlinear effect.
期刊介绍：
The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools.
Research Areas Include:
• Mathematical control theory
• Ordinary differential equations
• Partial differential equations
• Stochastic differential equations
• Topological dynamics
• Related topics
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