Harmonic Analysis and Nonlinear Hamiltonian Equations

调和分析和非线性哈密顿方程

• 批准号: 0621257
• 负责人: Andrew Comech
• 金额: $ 0.52万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2007-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0621257&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Nonlinear; Hamiltonian; Equations

The project suggests investigations in the area of Harmonic Analysis and Nonlinear Hamiltonian Equations. One direction of the research in the analysis of singular integral operators.Current interests include Lebesgue space estimates on Fourier integral operators associated to singular and nonsingular canonical relations. Such estimates provide important tools for the analysis of properties of solutions to nonlinear Hamiltonian systems: global existence, smoothness, and stability. The other direction of the research concerns specific properties of solitary wave solutions to nonlinear Hamiltonian systems.Such soliton-like solutions were proved to exist in manydifferent contexts. If the equation is invariant under theaction of some Lie group, then the solitary waves may possessstability properties. The program of research focuses on thestability of solitary waves with minimal energy, which areexpected to be of particular importance for applications.When the amplitude of the waves increases, their interaction with the medium becomes important. This interaction may seriouslychange the behavior of the waves and lead to the appearance ofnonlinear solitary waves, or solitons. Such nonlinear wavesdescribe central phenomena in fiber optics, plasma physics,theory of superconductivity, theory of elementary particles,meteorology, and oceanology. Detailed knowledge of propertiesof such waves, and in particular their stability or instability,will allow to predict the chances of sudden weather changes,develop optical waveguides, and describe quantum effects onthe scales being inexorably approached by today's electronicsand chip manufacturers, let alone the Experimental Physics.Natural phenomena pose new challenges to the intricately related fields, Harmonic Analysis and the Theory of Nonlinear Equations, and stimulate further refinement of the tools of modernMathematics.

该项目建议在谐波分析和非线性汉密尔顿方程的领域进行调查。 在分析奇异积分运算符的研究中，研究的一个方向包括对与单数和非义务规范关系相关的傅里​​叶积分运算符的Lebesgue空间估算。这样的估计为分析非线性哈密顿系统的属性分析提供了重要的工具：全球存在，平稳性和稳定性。研究的另一个方向是涉及非线性哈密顿系统的孤立波解决方案的特定特性。证明了类似孤子的溶液在许多不同的情况下存在。如果方程在某些谎言组的触发下不变，则孤立波可能具有稳定的特性。研究计划的重点是具有最小能量的孤立波的可疑性，这对应用而言特别重要。当波的振幅增加时，它们与介质的相互作用变得重要。这种相互作用可能会严重改变波浪的行为，并导致否则孤独波或孤子的出现。这种非线性波列在光纤，血浆物理学，超导性，基本颗粒理论，气象学和海洋学中的中心现象。 Detailed knowledge of propertiesof such waves, and in particular their stability or instability,will allow to predict the chances of sudden weather changes,develop optical waveguides, and describe quantum effects onthe scales being inexorably approached by today's electronicsand chip manufacturers, let alone the Experimental Physics.Natural phenomena pose new challenges to the intricately related fields, Harmonic Analysis and the Theory of Nonlinear Equations, and stimulate further refinement of现代音乐的工具。