Uniqueness of excited states to −Δu + u−u3 = 0 in three dimensions

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-07-19 DOI:10.2140/apde.2024.17.1887

Alex Cohen, Zhenhao Li, Wilhelm Schlag

{"title":"Uniqueness of excited states to −Δu + u−u3 = 0 in three dimensions","authors":"Alex Cohen, Zhenhao Li, Wilhelm Schlag","doi":"10.2140/apde.2024.17.1887","DOIUrl":null,"url":null,"abstract":"<p>We prove the uniqueness of several excited states to the ODE <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ÿ</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo>\n<mo>+</mo>\n<mo stretchy=\"false\">(</mo><mn>2</mn><mo>∕</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mi>ẏ</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo>\n<mo>+</mo>\n<mi>f</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">(</mo><mi>t</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <mn>0</mn></math>, <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>y</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo>\n<mo>=</mo>\n<mi>b</mi></math>, and <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ẏ</mi><mo stretchy=\"false\">(</mo><mn>0</mn><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <mn>0</mn></math>, for the model nonlinearity <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>f</mi><mo stretchy=\"false\">(</mo><mi>y</mi><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup>\n<mo>−</mo>\n<mi>y</mi></math>. The <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math>-th excited state is a solution with exactly <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi></math> zeros and which tends to <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mn>0</mn></math> as <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>t</mi>\n<mo>→</mo><mi>∞</mi></math>. These represent all smooth radial nonzero solutions to the PDE <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi mathvariant=\"normal\">Δ</mi><mi>u</mi>\n<mo>+</mo>\n<mi>f</mi><mo stretchy=\"false\">(</mo><mi>u</mi><mo stretchy=\"false\">)</mo>\n<mo>=</mo> <mn>0</mn></math> in <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math>. We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions. More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically. </p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.2140/apde.2024.17.1887","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}

引用次数: 0

Abstract

We prove the uniqueness of several excited states to the ODE $ÿ (t) + (2 ∕ t) ẏ (t) + f (y (t)) = 0$ , $y (0) = b$ , and $ẏ (0) = 0$ , for the model nonlinearity $f (y) = y^{3} - y$ . The $n$ -th excited state is a solution with exactly $n$ zeros and which tends to $0$ as $t \to \infty$ . These represent all smooth radial nonzero solutions to the PDE $Δ u + f (u) = 0$ in $H^{1}$ . We interpret the ODE as a damped oscillator governed by a double-well potential, and the result is proved via rigorous numerical analysis of the energy and variation of the solutions. More specifically, the problem of uniqueness can be formulated entirely in terms of inequalities on the solutions and their variation, and these inequalities can be verified numerically.

查看原文本刊更多论文

三维激发态对-Δu + u-u3 = 0的唯一性

我们证明了模型非线性 f(y)= y3-y 的 ODE ÿ(t)+(2∕t)ẏ(t)+f(y(t))=0、y(0)=b 和ẏ(0)=0 的几个激发态的唯一性。第 n 个激发态是一个恰好有 n 个零的解，随着 t→∞ 趋于 0。这些表示 H1 中 PDE Δu+f(u)= 0 的所有光滑径向非零解。我们将该 ODE 解释为受双井势能支配的阻尼振荡器，并通过对解的能量和变化进行严格的数值分析来证明结果。更具体地说，唯一性问题完全可以用解及其变化的不等式来表述，而且这些不等式可以用数值来验证。

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分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.