Solutions of Gross–Pitaevskii equation with periodic potential in dimension three

IF 0.7 4区数学 Q2 MATHEMATICS St Petersburg Mathematical Journal Pub Date : 2024-04-12 DOI:10.1090/spmj/1798

Yu. Karpeshina, Seonguk Kim, R. Shterenberg

{"title":"Solutions of Gross–Pitaevskii equation with periodic potential in dimension three","authors":"Yu. Karpeshina, Seonguk Kim, R. Shterenberg","doi":"10.1090/spmj/1798","DOIUrl":null,"url":null,"abstract":"<p>Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script upper G subset-of double-struck upper R cubed\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"script\">G</mml:mi> </mml:mrow> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathcal {G}\\subset \\mathbb {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that for every <inline-formula content-type=\"math/tex\"> <tex-math> \\vv k\\in \\mathcal {G}</tex-math></inline-formula> there is a solution asymptotically close to a plane wave <inline-formula content-type=\"math/tex\"> <tex-math> Ae^{i\\langle \\vv {k},\\vv {x}\\rangle }</tex-math></inline-formula> as <inline-formula content-type=\"math/tex\"> <tex-math> |\\vv k|\\to \\infty </tex-math></inline-formula>, given <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently small.</p>","PeriodicalId":51162,"journal":{"name":"St Petersburg Mathematical Journal","volume":"13 2 1","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"St Petersburg Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/spmj/1798","RegionNum":4,"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

Abstract

Quasiperiodic solutions of the Gross–Pitaevskii equation with a periodic potential in dimension three are studied. It is proved that there is an extensive “nonresonant” set G ⊂ R 3 \mathcal {G}\subset \mathbb {R}^3 such that for every \vv k\in \mathcal {G} there is a solution asymptotically close to a plane wave Ae^{i\langle \vv {k},\vv {x}\rangle } as |\vv k|\to \infty , given A A is sufficiently small.

查看原文

微信好友朋友圈 QQ好友复制链接

本刊更多论文

具有周期势能的格罗斯-皮塔耶夫斯基方程三维解

研究了三维中具有周期势的格罗斯-皮塔耶夫斯基方程的准周期解。研究证明，存在一个广泛的 "非共振 "集合 G ⊂ R 3 \mathcal {G}\subset \mathbb {R}^3 ，这样对于 \mathcal {G} 中的每一个 \vv k\ 都有一个近似接近于平面波 Ae^{i\langle \vv {k},\vv {x}\rangle } 的解，因为 |\vv k|\to \infty ，给定 A A 足够小。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文去求助

来源期刊

St Petersburg Mathematical Journal MATHEMATICS-

CiteScore

1.00

自引率

12.50%

发文量

审稿时长

>12 weeks

期刊介绍： This journal is a cover-to-cover translation into English of Algebra i Analiz, published six times a year by the mathematics section of the Russian Academy of Sciences.

期刊最新文献

Shape, velocity, and exact controllability for the wave equation on a graph with cycle On Kitaev’s determinant formula Resolvent stochastic processes Complete nonselfadjointness for Schrödinger operators on the semi-axis Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model