{"title":"Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations","authors":"Joackim Bernier, Erwan Faou, Benoît Grébert","doi":"10.1007/s40818-020-00089-5","DOIUrl":null,"url":null,"abstract":"<div><p>We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant <i>M</i> and a sufficiently small parameter <span>\\(\\varepsilon \\)</span>, for generic initial data of size <span>\\(\\varepsilon \\)</span>, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order <span>\\(\\varepsilon ^{M+1}\\)</span>. This implies that for such initial data <i>u</i>(0) we control the Sobolev norm of the solution <i>u</i>(<i>t</i>) for time of order <span>\\(\\varepsilon ^{-M}\\)</span>. Furthermore this property is locally stable: if <i>v</i>(0) is sufficiently close to <i>u</i>(0) (of order <span>\\(\\varepsilon ^{3/2}\\)</span>) then the solution <i>v</i>(<i>t</i>) is also controled for time of order <span>\\(\\varepsilon ^{-M}\\)</span>.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":null,"pages":null},"PeriodicalIF":2.4000,"publicationDate":"2020-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40818-020-00089-5","citationCount":"30","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annals of Pde","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40818-020-00089-5","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 30
Abstract
We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant M and a sufficiently small parameter \(\varepsilon \), for generic initial data of size \(\varepsilon \), the flow is conjugated to an integrable flow up to an arbitrary small remainder of order \(\varepsilon ^{M+1}\). This implies that for such initial data u(0) we control the Sobolev norm of the solution u(t) for time of order \(\varepsilon ^{-M}\). Furthermore this property is locally stable: if v(0) is sufficiently close to u(0) (of order \(\varepsilon ^{3/2}\)) then the solution v(t) is also controled for time of order \(\varepsilon ^{-M}\).