亚声速区Gross-Pitaevskii方程的有限能量行波

IF 1.7 1区数学 Q1 MATHEMATICS

American Journal of Mathematics Pub Date : 2019-11-07 DOI:10.1353/ajm.2023.0002

J. Bellazzini, D. Ruiz

{"title":"亚声速区Gross-Pitaevskii方程的有限能量行波","authors":"J. Bellazzini, D. Ruiz","doi":"10.1353/ajm.2023.0002","DOIUrl":null,"url":null,"abstract":"Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mari\\\\c\\{s\\} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Mari\\\\c\\{s\\}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":"20 1","pages":"109 - 149"},"PeriodicalIF":1.7000,"publicationDate":"2019-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"13","resultStr":"{\"title\":\"Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime\",\"authors\":\"J. Bellazzini, D. Ruiz\",\"doi\":\"10.1353/ajm.2023.0002\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mari\\\\\\\\c\\\\{s\\\\} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Mari\\\\\\\\c\\\\{s\\\\}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.\",\"PeriodicalId\":7453,\"journal\":{\"name\":\"American Journal of Mathematics\",\"volume\":\"20 1\",\"pages\":\"109 - 149\"},\"PeriodicalIF\":1.7000,\"publicationDate\":\"2019-11-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"13\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"American Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1353/ajm.2023.0002\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"American Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1353/ajm.2023.0002","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: 13

摘要

摘要:本文研究了Gross-Pitaevskii方程有限能量行波的存在性。这个问题在文献中得到了很多关注，但是在整个亚音速范围内解的存在性一直是一个开放的问题，直到2013年Mari\\c\{s\}的工作。然而，这样的结果只在维度3及更高的维度上有效。本文首先证明了在亚音速范围内几乎每一个速度值都存在有限能量行波。我们的论证在二维和三维中同样有效。有了这个结果，紧凑性论证可以填补可接受的速度范围。我们可以在维度3中这样做，通过Mari\\c\{s\}恢复上述结果。平面情况更为复杂，紧性论证仅在对近似解的涡集附加假设的情况下才成立。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Finite energy traveling waves for the Gross-Pitaevskii equation in the subsonic regime

Abstract:In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem hasdeserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Mari\\c\{s\} in 2013. However, such result is valid only in dimension 3 and higher. In this paperwe first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3.With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3,recovering the aforementioned result by Mari\\c\{s\}. The planar case turns out to be more intricate and the compactness argumentworks only under an additional assumption on the vortex set of the approximating solutions.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

American Journal of Mathematics 数学-数学

CiteScore

3.20

自引率

0.00%

发文量

审稿时长

24 months

期刊介绍： The oldest mathematics journal in the Western Hemisphere in continuous publication, the American Journal of Mathematics ranks as one of the most respected and celebrated journals in its field. Published since 1878, the Journal has earned its reputation by presenting pioneering mathematical papers. It does not specialize, but instead publishes articles of broad appeal covering the major areas of contemporary mathematics. The American Journal of Mathematics is used as a basic reference work in academic libraries, both in the United States and abroad.