Singular and Spatially Heterogeneous Perturbations of Solitary Waves

孤立波的奇异和空间异质扰动

• 批准号: 2006172
• 负责人: Jay Wright
• 金额: $ 16.5万
• 依托单位: Drexel University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-09-01 至 2024-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006172&HistoricalAwards=false
• 关键词: Singular; Spatially; Heterogeneous; Perturbations; Solitary

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.

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，虽然有很大的不同，但从数学的角度来看，它们有着深刻的共性。所有这些系统的建模、分析和计算机模拟的技术水平是非常复杂的。尽管如此，一些看似简单的、与物理相关的效应，却使系统的数学分析方式变得极其复杂，使目前的方法显得不足。该项目旨在推进波的数学处理，以纳入这类物理效应的两大类。其中第一个是空间的，也就是说，与系统在其范围内不均匀这一事实有关的影响。例如，这种非均质性产生于包括水波模型中的底部地形、弹性材料中的层压或工程材料中的结构变异性。第二种是所谓的奇异效应。这些包括计算流体中的表面张力，在等离子体模型中加入多种离子，以及引入分子链中的缺陷。该项目旨在(a)为这些影响设计模型，(b)开发数学上严谨、广泛适用且高度准确的定量描述，以及(c)实现用于此类系统模拟的新算法。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。空间异质性破坏了平移不变性，平移不变性是行波存在的重要因素。奇异效应是出了名的难以预测。然而，结合这些影响并不需要完全消除连贯的结构。例如，可能是一个行波变成了一个纳米粒子，也就是说，一个行波是一个局部孤立核和一个非常小振幅周期波的叠加。在重力-毛细波问题和各种哈密顿晶格模型中已经发现了纳米粒子的存在。另一种可能性是，孤立波在极长时间内看似不变地传播，但最终退化为色散波。也就是说，孤立波转变为长寿命的瞬态或亚稳态解。该项目的主要目标是：确定具有纳米粒子溶液的系统并严格确定其存在；推进严格的纳米粒子理论，使其更具描述性，并在孤立核较大的情况下工作；为哈密顿偏微分方程和格微分方程设计并实现高阶辛积分器，以模拟超长时间尺度上的亚稳态孤立波；严格地证明了亚稳态孤立波的存在；并通过分析和模拟，确定与孤波的空间异质或奇异扰动相关的新现象学。该奖项反映了美国国家科学基金会的法定使命，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，被认为值得支持。

项目成果

Using random walks to establish wavelike behavior in a linear FPUT system with random coefficients

使用随机游走在具有随机系数的线性 FPUT 系统中建立波状行为

• DOI: 10.3934/dcdss.2021100
• 发表时间: 2022
• 期刊: Discrete and Continuous Dynamical Systems - S
• 作者: McGinnis, Joshua A.;Wright, J. Douglas
• 通讯作者: Wright, J. Douglas

A simple model of radiating solitary waves

辐射孤立波的简单模型

• DOI: 10.1016/j.wavemoti.2022.102971
• 发表时间: 2022
• 期刊: Wave Motion
• 影响因子: 2.4
• 作者: Wright, J. Douglas

Well-Posedness and Asymptotics of a Coordinate-Free Model of Flame Fronts

火焰锋面无坐标模型的适定性和渐近性

• DOI: 10.1137/20m1370793
• 发表时间: 2021
• 期刊: SIAM Journal on Applied Dynamical Systems
• 影响因子: 2.1
• 作者: Ambrose, David M.;Hadadifard, Fazel;Wright, J. Douglas
• 通讯作者: Wright, J. Douglas

Mass‐in‐mass lattices with small internal resonators

具有小型内部谐振器的质量-质量晶格

• DOI: 10.1111/sapm.12340
• 发表时间: 2021
• 期刊: Studies in Applied Mathematics
• 影响因子: 2.7
• 作者: Hadadifard, Fazel;Wright, J. Douglas
• 通讯作者: Wright, J. Douglas

Solitary waves in mass-in-mass lattices

质中晶格中的孤立波

• DOI: 10.1007/s00033-020-01384-8
• 发表时间: 2020
• 期刊: Zeitschrift für angewandte Mathematik und Physik
• 作者: Faver, Timothy E.;Goodman, Roy H.;Wright, J. Douglas
• 通讯作者: Wright, J. Douglas