{"title":"无界扰动偏微分方程的退化 KAM 定理","authors":"Mei Na Gao, Jian Jun Liu","doi":"10.1007/s10114-024-3159-1","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper, an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi, Berti, Magistrelli, and us. As applications, for derivative nonlinear Schrödinger equation with periodic boundary conditions, quasi-periodic solutions around constant solutions are obtained.</p></div>","PeriodicalId":50893,"journal":{"name":"Acta Mathematica Sinica-English Series","volume":"40 11","pages":"2719 - 2734"},"PeriodicalIF":0.8000,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Degenerate KAM Theorem for Partial Differential Equations with Unbounded Perturbations\",\"authors\":\"Mei Na Gao, Jian Jun Liu\",\"doi\":\"10.1007/s10114-024-3159-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this paper, an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi, Berti, Magistrelli, and us. As applications, for derivative nonlinear Schrödinger equation with periodic boundary conditions, quasi-periodic solutions around constant solutions are obtained.</p></div>\",\"PeriodicalId\":50893,\"journal\":{\"name\":\"Acta Mathematica Sinica-English Series\",\"volume\":\"40 11\",\"pages\":\"2719 - 2734\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-11-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Sinica-English Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10114-024-3159-1\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Sinica-English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10114-024-3159-1","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
本文在定性非退化条件下，建立了具有无界扰动和双法频的无限维 KAM 定理。这是 Bambusi、Berti、Magistrelli 和我们对有界扰动的退化 KAM 定理的扩展。作为应用，对于具有周期性边界条件的导数非线性薛定谔方程，可以得到恒定解周围的准周期解。
A Degenerate KAM Theorem for Partial Differential Equations with Unbounded Perturbations
In this paper, an infinite dimensional KAM theorem with unbounded perturbations and double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with bounded perturbations by Bambusi, Berti, Magistrelli, and us. As applications, for derivative nonlinear Schrödinger equation with periodic boundary conditions, quasi-periodic solutions around constant solutions are obtained.
期刊介绍：
Acta Mathematica Sinica, established by the Chinese Mathematical Society in 1936, is the first and the best mathematical journal in China. In 1985, Acta Mathematica Sinica is divided into English Series and Chinese Series. The English Series is a monthly journal, publishing significant research papers from all branches of pure and applied mathematics. It provides authoritative reviews of current developments in mathematical research. Contributions are invited from researchers from all over the world.
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