Degenerate dispersive effects in partial and lattice differential equations

偏微分方程和晶格微分方程中的简并色散效应

• 批准号: 1105635
• 负责人: Jay Wright
• 金额: $ 20.28万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105635&HistoricalAwards=false
• 关键词: Degenerate; dispersive; effects; partial; lattice

A spatially extended system is said to be dispersive when the speed of propagation of a wave depends upon that wave's frequency. Linear dispersive effects play a fundamental role in the study of a large number of physical scenarios and there has been an explosion of results concerning semi-linear dispersive equations. Nevertheless there are situations in which the speed of propagation of a wave depends on the wave's frequency as well as its amplitude. That is to say the mechanism which generates dispersion is nonlinear. It is the purpose of this project to develop mathematically rigorous theory for such equations when the nonlinear dispersive effects are degenerate (i.e., they vanish very rapidly as the amplitude tends to zero). The degeneracy of the equations allows for the existence of classical solutions which are not smooth and there is substantial numerical and formal evidence that cases arise in which degenerate dispersion leads to catastrophic instability akin to that of a backwards heat equation. Consequently, one of the principal goals of the project is to develop the existence theory for degenerate dispersive partial differential equations.Degenerate dispersive equations arise as models for the dynamics of chains and lattices of hard spheres. Experimental results demonstrate that such chains exhibit tightly focused pulses which are easily generated and highly tunable. These narrow pulses are expected to find uses in non-destructive testing, shock absorption, remote sensing and medical imaging. At this time, there is no theoretically rigorous explanation for the stability and robust nature of these pulses. Though there are results concerning similar physical problems where the dispersive effects are not degenerate, they rely strongly on the fact that slow moving pulses are very wide and very small in amplitude. In the case of hard spheres, the interesting pulse solutions have a fixed width, independent of speed and amplitude and it is not obvious how to extend the known results to this case. A primary goal of the project is developing mathematical tools for the study of the stability and robustness of such pulses. The inclusion and training of graduate and undergraduate students is an integral part of this project. Among other projects, undergraduate students will be involved into investigation of an important application, models for traffic flow.

当波的传播速度取决于该波的频率时，空间扩展的系统被称为色散的。线性色散效应在大量物理场景的研究中起着基础性的作用，并且关于半线性色散方程的结果已经出现了爆炸式的增长。 然而，在某些情况下，波的传播速度取决于波的频率和振幅。 也就是说，产生色散的机制是非线性的。 本项目的目的是在非线性色散效应退化（即，当振幅趋于零时，它们非常迅速地消失）。方程的退化允许经典的解决方案，这是不光滑的存在，有大量的数值和正式的证据表明，出现的情况下，退化的色散导致灾难性的不稳定类似于向后热方程。 因此，该项目的主要目标之一是发展退化色散偏微分方程的存在性理论。退化色散方程作为硬球链和晶格动力学的模型出现。 实验结果表明，这种链表现出紧密聚焦的脉冲，这是很容易产生和高度可调。这些窄脉冲有望用于无损检测、减震、遥感和医学成像。 在这个时候，没有理论上严格的解释这些脉冲的稳定性和鲁棒性。虽然也有关于类似的物理问题的色散效应不退化的结果，他们强烈依赖于这样一个事实，即缓慢移动的脉冲是非常广泛的，非常小的振幅。 在硬球的情况下，有趣的脉冲解具有固定的宽度，与速度和幅度无关，并且如何将已知的结果扩展到这种情况并不明显。 该项目的主要目标是开发数学工具，用于研究这种脉冲的稳定性和鲁棒性。对研究生和本科生的包容和培训是该项目的一个组成部分。在其他项目中，本科生将参与调查一个重要的应用，交通流模型。