Linear and Nonlinear Waves

线性和非线性波

• 批准号: 9706780
• 负责人: Avraham Soffer
• 金额: $ 8.35万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706780&HistoricalAwards=false
• 关键词: Linear; Nonlinear; Waves

9706780 Soffer In this project the analysis of large time behavior of a class of nonhomogeneous and nonlinear wave equations is pursued. This class include solitary wave solutions interacting with radiation. The aim is to develop the scattering theory for nonintegrable multichannel problems. The Klein-Gordon equation with potential term and nonlinearity is one case to be considered. The other example is the nonlinear Schrodinger equation with a potential holding more than one bound state. Other parts of the proposal deal with the spectral theory of N-body dispersive equations, in particular the proof of local decay and absence of singular continuous spectrum for such hamiltonians. The project deals with the large time behavior of radiation waves and waves in media. Analytic methods to find the radiation of molecules, spontaneous or stimulated, are developed. The nature of this radiation is used in various applications to determine the structure and chemical properties of the source. The second part of the project deals with waves travelling through a medium. Such waves often change the medium and thus affect the way the wave moves. Such self-influence lead to NONLINEAR wave equations with new and complicated behavior not seen in vacuum. The most striking examples are singularities that form in a finite time. These are used to explain diverse phenomena, from black hole formation to air and plasma turbulence. Another phenomenon is the formation of "clumps" of waves that move in the medium with unusually high stability, like solitons in optic fibers. The current project is devoted to the analysis of the large time radiation and interaction of nonlinear waves and solitons. This includes the study of the effect of defects and impurities in optic fibers on the motion and structure of optical solitons used in high speed communications.

本课题研究了一类非齐次非线性波动方程的大时间行为。本课程包括与辐射相互作用的孤波解。目的是发展不可积多通道问题的散射理论。具有位项和非线性的Klein-Gordon方程是需要考虑的一种情况。另一个例子是非线性薛定谔方程，它具有一个以上的束缚态。论文的其他部分讨论了n体色散方程的谱理论，特别是证明了这类哈密顿量的局部衰减和奇异连续谱的不存在。该项目研究辐射波和介质中的波的大时间特性。发展了发现分子自发或受激辐射的解析方法。这种辐射的性质在各种应用中被用来确定源的结构和化学性质。这个项目的第二部分是关于波在介质中的传播。这种波经常改变介质，从而影响波的运动方式。这种自影响导致非线性波动方程具有真空中没有的新的复杂行为。最显著的例子是在有限时间内形成的奇点。这些理论被用来解释各种现象，从黑洞形成到空气和等离子体湍流。另一种现象是形成“团块”波，这些波在介质中以异常高的稳定性移动，就像光纤中的孤子一样。本课题主要研究非线性波与孤子的大时间辐射与相互作用。这包括研究光纤中的缺陷和杂质对用于高速通信的光孤子的运动和结构的影响。