A comprehensive study of nonlinear problems via variational approaches

通过变分方法综合研究非线性问题

• 批准号: 20340037
• 负责人: TANAKA Kazunaga
• 金额: $ 11.32万
• 依托单位: Waseda University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2008
• 资助国家: 日本
• 起止时间: 2008 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-20340037/
• 关键词: 変分問題; 非線型楕円型方程式; 特異摂動問題; ハミルトン系; 変分法; 非線形楕円型方程式

We study nonlinear problems via variational approaches. Especially (1) we study singular perturbation problems for nonlinear Schrodinger equations and systems. We introduce a new purely variational method which enables us to construct concentrating solutions in a very general setting. (2) We study nonlinear elliptic equations and systems in various settings. We give a new variational construction of radially symmetric ground states. We also study stability and instability of solutions. (3) We also study highly oscillatory solutions in 1-dimensional singular perturbation problems. We give characterization and existence result.

我们用变分方法研究非线性问题。特别是(1)研究了非线性薛定谔方程和系统的奇摄动问题。我们引入了一种新的纯变分方法，它使我们能够在非常一般的环境下构造集中解。(2)研究了不同背景下的非线性椭圆型方程和方程组。给出了径向对称基态的一种新的变分结构。我们还研究了解的稳定性和不稳定性。(3)我们还研究了一维奇异摄动问题的高振动解。给出了刻画和存在的结果。

项目成果

A new variational approach for a singularly perturbed nonlinear Schrodinger equaiton

奇异摄动非线性薛定谔方程的新变分法

• 发表时间: 2011
• 作者: T.Okuda;J.Penedones;二森都;S.Aoki;K.Tanaka
• 通讯作者: K.Tanaka

Inequalities associated with dilations

与扩张相关的不平等

• 发表时间: 2009
• 期刊: Commun. Contemporary Math. 11
• 作者: H. Nakao;S. Kodama;K. Kiyoto;D. Bizen;Y. murakami;et. al.;T. Ozawa
• 通讯作者: T. Ozawa

Standing Waves for Nonlinear Schrödinger Equations with a General Nonlinearity: One and Two Dimensional Cases

• DOI: 10.1080/03605300701518174
• 发表时间: 2008-05
• 期刊: Communications in Partial Differential Equations
• 影响因子: 1.9
• 作者: Jaeyoung Byeon;L. Jeanjean;Kazunaga Tanaka

Solutions with clustering peaks for a singularly perturbed NLS

奇异扰动 NLS 的聚类峰解决方案

• 发表时间: 2011
• 作者: T. Okuda;S. Iguchi;K. Kohno;赤堀次郎;K. Tanaka
• 通讯作者: K. Tanaka

Semiclassical Standing Waves With Clustering Peaks for Nonlinear Schrodinger Equations

• DOI: 10.1090/memo/1076
• 发表时间: 2014-05
• 作者: Jaeyoung Byeon;Kazunaga Tanaka