Stability of waves in discrete and continuous dynamical systems

离散和连续动力系统中波的稳定性

• 批准号: 1313107
• 负责人: Atanas Stefanov
• 金额: $ 18.47万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1313107&HistoricalAwards=false
• 关键词: Stability; waves; discrete; continuous; dynamical

The main theme of the proposed research will be the linear and asymptotic stability of special solutions (waves) of a wide class of partial differential equations. In a series of recent works, the PI and collaborators have characterized the linear stability of travelling and standing waves for second order in time equations and systems. Such results provide necessary information for the asymptotic stability of the same waves. The PI and collaborators will build on their previous work to show asymptotic stability of standing waves for the following models: the Klein-Gordon equation, the sine Gordon equation and the Dirac equation. A second goal of the proposal is to study the existence and stability of coherent structures, arising in spatially discrete models. More concretely, the project will deal, among other things, with Hertzian granular chains/crystals and the discrete nonlinear Schr\"odinger equation.The project deals with nonlinear dispersive equations, which model wavelike behavior of important physical processes. Important class of problems under consideration include the propagation of light in optical waveguides, motion of quantum particles, the mechanics of fluids to mention a few. The overarching theme of the investigation will be the stability of coherent configuration - that is, if one is initially close to such coherent structure, does it stay close to it forever? The mathematical formulation of such problems, as well as their analysis and predictions about their long time behavior will greatly enhance our understanding of these and related processes.

拟议的研究的主要主题是特殊溶液（波）的线性和渐近稳定性（WAVE）。在最近的一系列作品中，PI和合作者表征了在时间方程式和系统中二阶旅行和站立波的线性稳定性。 这样的结果为相同波的渐近稳定性提供了必要的信息。 PI和合作者将基于他们以前的工作，以显示以下模型的渐进挥手稳定性：Klein-Gordon方程，正弦Gordon方程和DIRAC方程。该提案的第二个目标是研究在空间离散模型中产生的相干结构的存在和稳定性。更具体地说，该项目将与赫兹的颗粒链/晶体和离散的非线性schr \“ odinger”方程式处理。该项目涉及非线性分散方程，这些方程模拟了重要的物理过程的型号，这些方程模拟了重要的物理过程。重要的问题类别的重要类别包括在量化波动中的重要性，包括量化的主题，这些机构的机制是一定的，这些机制的机制是一定的，这些机制是一定的。研究将是连贯配置的稳定性 - 也就是说，如果最初接近这种相干结构，它是否会永远与之保持联系？