Singular and Spatially Heterogeneous Perturbations of Solitary Waves

尽管有很大的不同，但从数学的角度来看，海洋表面的波浪、长分子的振动、过热等离子体的振荡、电力系统中的电力动力学以及工程材料的运动都有很深的共性。所有这些系统的建模、分析和计算机模拟的最新技术水平是非常复杂的。尽管如此，有一些看似简单、与物理相关的效应，却使系统的数学分析方式变得复杂，并使当前的方法变得不足。该项目旨在推进波的数学处理，以纳入两大类这样的物理效应。第一个是空间效应，即与系统在其范围内不均匀的事实有关的效应。例如，这种不均匀性来自于水波模型中的底部地形、弹性体中的层压或工程材料中的结构变化。第二种是所谓的奇异效应。这些包括考虑流体中的表面张力，在等离子体模型中引入多种离子，以及在分子链中引入缺陷。该项目的目的是（a）设计这些影响的模型，（B）开发数学上严格的、广泛适用的和高度准确的定量描述，（c）实现模拟这种系统的新算法。该项目将为本科生和研究生提供研究培训机会。从技术上讲，目标是了解上述现象如何影响非线性、双曲和/或色散微分方程中相干结构的存在性、稳定性和（特别是）动力学。空间不均匀性破坏了平移不变性，而平移不变性是行波存在的一个基本要素。奇异效应是出了名的不可预测。然而，纳入这些效应并不需要完全消除相干结构。例如，可能的是，行波变成纳米鳍，也就是说，行波是局部孤立核和非常小振幅的周期波的叠加。众所周知，纳米鳍存在于重力毛细波问题和各种汉密尔顿晶格模型中。另一种可能性是孤立波在很长一段时间内似乎没有变化，但最终会恶化为色散波。也就是说，孤立波转变成一个长寿命的瞬态，或亚稳态，解决方案。该项目的主要目标是：确定具有纳米鳍解决方案的系统，并严格确定其存在性;推进严格的纳米鳍理论，使其更具描述性，并在孤核较大的环境中发挥作用;设计和实施哈密顿偏微分方程和晶格微分方程的高阶辛积分器，以模拟很长时间尺度上的亚稳态孤波;严格证明亚稳态孤立波的存在;通过分析和模拟识别与孤立波的空间异质性或奇异扰动相关的新现象。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估而被认为值得支持。

Although quite different, waves on the surface of the ocean, vibrations in a long molecule, oscillations in a superheated plasma, the dynamics of electricity in power systems, and the motion of engineered materials have deep commonalities from a mathematical point of view. The state of the art for the modeling, analysis, and computer simulation of all these systems is very sophisticated. Nonetheless, there are seemingly simple, physically-relevant effects that profoundly complicate the way in which the systems are analyzed mathematically and that make current methods insufficient. This project aims to advance the mathematical treatment of waves to incorporate two broad classes of such physical effects. The first of these is spatial, that is, the effects related to the fact that a system is not uniform throughout its extent. For example, such heterogeneity arises from including bottom topography in water wave models, lamination in elastica, or structural variability in engineered materials. The second are so-called singular effects. These include accounting for surface tension in fluids, incorporating multiple species of ions in plasma models, and introduction of defects in molecular chains. The project aims to (a) devise models for these effects, (b) develop mathematically rigorous, broadly applicable, and highly accurate quantitative descriptions, and (c) implement novel algorithms for the simulation of such systems. The project will provide research training opportunities for undergraduate and graduate students. More technically, the goal is to understand how the above phenomena affect the existence, stability, and (especially) dynamics of coherent structures in nonlinear, hyperbolic and/or dispersive differential equations. Spatial heterogeneity ruins translation invariance, an essential ingredient for the existence of traveling waves. Singular effects are famously unpredictable. Nevertheless, incorporating these sorts of effects need not eliminate entirely the coherent structures. For instance, it may be that a traveling wave becomes a nanopteron, which is to say a traveling wave that is the superposition of a localized solitary core and a very small amplitude periodic wave. Nanopterons are already known to exist in the gravity-capillary wave problem and a variety of Hamiltonian lattice models. Another possibility is that a solitary wave propagates seemingly unchanged for an extremely long time but eventually deteriorates into dispersive waves. That is to say, the solitary wave transforms into a long-lived transient, or metastable, solution. This project’s principal aims are: identify systems that possess nanopteron solutions and establish their existence rigorously; advance the rigorous nanopteron theory to be more descriptive and to work in settings where the solitary core is large; devise and implement high-order symplectic integrators for Hamiltonian partial and lattice differential equations to simulate metastable solitary waves over very long time scales; prove rigorously the existence of metastable solitary waves; and identify new phenomenology related to spatially heterogeneous or singular perturbations of solitary waves through analysis and simulation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.