{"title":"论立方非线性薛定谔方程某些椭圆解的存在性","authors":"H. W. Schürmann, V. S. Serov","doi":"10.1134/S0040577924040044","DOIUrl":null,"url":null,"abstract":"<p> We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form <span>\\(\\Psi(t,z)=(f(t,z)+id(z))e^{i\\phi(z)}\\)</span> with <span>\\(f,\\phi,d\\in\\mathbb{R}\\)</span>, we prove that they are nonexistent in the general case <span>\\(f_z\\neq 0\\)</span>, <span>\\(f_t\\neq 0\\)</span>, <span>\\(d_z\\neq 0\\)</span>. In the three nongeneric cases (<span>\\(f_z\\neq 0\\)</span>), (<span>\\(f_t\\neq 0\\)</span>, <span>\\(f_t=0\\)</span>, <span>\\(d_z=0\\)</span>), and (<span>\\(f_z=0\\)</span>, <span>\\(f_t\\neq 0\\)</span>), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":"219 1","pages":"557 - 566"},"PeriodicalIF":1.0000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the existence of certain elliptic solutions of the cubically nonlinear Schrödinger equation\",\"authors\":\"H. W. Schürmann, V. S. Serov\",\"doi\":\"10.1134/S0040577924040044\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p> We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form <span>\\\\(\\\\Psi(t,z)=(f(t,z)+id(z))e^{i\\\\phi(z)}\\\\)</span> with <span>\\\\(f,\\\\phi,d\\\\in\\\\mathbb{R}\\\\)</span>, we prove that they are nonexistent in the general case <span>\\\\(f_z\\\\neq 0\\\\)</span>, <span>\\\\(f_t\\\\neq 0\\\\)</span>, <span>\\\\(d_z\\\\neq 0\\\\)</span>. In the three nongeneric cases (<span>\\\\(f_z\\\\neq 0\\\\)</span>), (<span>\\\\(f_t\\\\neq 0\\\\)</span>, <span>\\\\(f_t=0\\\\)</span>, <span>\\\\(d_z=0\\\\)</span>), and (<span>\\\\(f_z=0\\\\)</span>, <span>\\\\(f_t\\\\neq 0\\\\)</span>), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions. </p>\",\"PeriodicalId\":797,\"journal\":{\"name\":\"Theoretical and Mathematical Physics\",\"volume\":\"219 1\",\"pages\":\"557 - 566\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theoretical and Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://link.springer.com/article/10.1134/S0040577924040044\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical and Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://link.springer.com/article/10.1134/S0040577924040044","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
摘要
Abstract We consider solutions of the cubically nonlinear Schrödinger equation.对于一类形式为 \(\Psi(t,z)=(f(t,z)+id(z))e^{i\phi(z)}\) with \(f,\phi,d\in\mathbb{R}\) 的解,我们证明它们在一般情况下是不存在的((f_z\neq 0\ )、(f_t\neq 0\ )、(d_z\neq 0\ )。在三种非一般情况下((f_z\neq 0)),((f_t\neq 0),(f_t=0),(d_z=0)),和((f_z=0),(f_t\neq 0)),我们提出了一个双参数的解集,我们找到了指定实有界解和无界解的约束条件。
On the existence of certain elliptic solutions of the cubically nonlinear Schrödinger equation
We consider solutions of the cubically nonlinear Schrödinger equation. For a certain class of solutions of the form \(\Psi(t,z)=(f(t,z)+id(z))e^{i\phi(z)}\) with \(f,\phi,d\in\mathbb{R}\), we prove that they are nonexistent in the general case \(f_z\neq 0\), \(f_t\neq 0\), \(d_z\neq 0\). In the three nongeneric cases (\(f_z\neq 0\)), (\(f_t\neq 0\), \(f_t=0\), \(d_z=0\)), and (\(f_z=0\), \(f_t\neq 0\)), we present a two-parameter set of solutions, for which we find the constraints specifying real bounded and unbounded solutions.
期刊介绍:
Theoretical and Mathematical Physics covers quantum field theory and theory of elementary particles, fundamental problems of nuclear physics, many-body problems and statistical physics, nonrelativistic quantum mechanics, and basic problems of gravitation theory. Articles report on current developments in theoretical physics as well as related mathematical problems.
Theoretical and Mathematical Physics is published in collaboration with the Steklov Mathematical Institute of the Russian Academy of Sciences.