Stability of waves in discrete and continuous dynamical systems

所提出的研究的主要主题将是一大类偏微分方程解(波)的线性和渐近稳定性。在最近的一系列工作中，PI及其合作者刻画了时间方程和系统中二阶行波和驻波的线性稳定性。这些结果为相同波的渐近稳定性提供了必要的信息。PI和合作者将在他们之前的工作的基础上，证明下列模型的驻波的渐近稳定性：Klein-Gordon方程，Sine Gordon方程和Dirac方程。该提议的第二个目标是研究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，该项目将处理赫兹颗粒链/晶体和离散的非线性薛定谔方程。该项目处理非线性色散方程，它模拟重要物理过程的波动行为。正在考虑的重要问题包括光在光波导中的传播，量子粒子的运动，流体的力学等等。调查的首要主题将是连贯结构的稳定性--也就是说，如果一开始接近这种连贯结构，它会永远保持接近吗？对这类问题的数学表述，以及它们对其长期行为的分析和预测，将极大地增强我们对这些过程和相关过程的理解。

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.