Collaborative Research: Stability of Nonlinear Wave Structures in Lattices

这个合作研究项目的主要研究人员研究了晶格中非线性波的稳定性。在许多物理和生物系统中，从机械超材料和纳米机电器件到生物聚合物，空间离散结构和非线性的相互作用经常导致非线性波形的形成，当它们在系统中传播时，这些波形要么相干地传输要么释放能量。特别是，最近关于颗粒材料的实验和理论工作已经提高了对各种此类结构的存在条件和性质的理解，包括行脉冲和冲击波。然而，人们对这些结构稳定的条件知之甚少。该项目的目标是在一个框架中获得这样的标准，该框架将早期的工作扩展到更一般的波类，并扩展到更广泛的设置，以解释更高维度、晶格结构、外部驱动和阻尼的影响。对非线性波稳定性准则的基本理解在许多不同的领域都很重要，包括材料科学、凝聚态物理、机械工程和生物物理学。这项工作提供的见解可以帮助设计用于能量通道，减震和减振的新设备。这个合作项目的一个组成部分是教学和训练跨学科研究领域的非线性波格现象的研究生。为此，研究生参与研究。本项目建立在主要研究人员正在进行的关于一维晶格中孤立行波的稳定性及其与离散呼吸体稳定性的密切联系的联合工作的基础上。这包括开发新的分析稳定性标准，以及将该框架扩展到更广泛的设置所需的最先进的数值方法，包括二维和非均质晶格，同时考虑更一般的行波类别。特别是，具有振荡尾的波形的稳定性，如共振颗粒链中的纳米tera和位错模型中的广义扭结，进行了研究。该项目包括研究不稳定的动力学含义，并检查耗散、远程相互作用和外部驱动的影响。该项目的目标还包括澄清与孤立行波相关的线性谱（及其在不稳定临界点的变化）与相应的离散呼吸弗洛凯谱的关系，以及将此类波的稳定性标准与连续统系统的既定功能分析框架联系起来。研究生参与研究。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。

The principal investigators of this collaborative research project study the stability of nonlinear waves in lattices. The interplay of spatially discrete structure and nonlinearity in many physical and biological systems, from mechanical metamaterials and nanoelectromechanical devices to biopolymers, often results in formation of nonlinear waveforms that either coherently transport or release energy as they propagate through the system. In particular, recent experimental and theoretical work on granular materials has advanced the understanding of the conditions for existence and properties of a variety of such structures, including traveling pulses and shock waves. However, much less is known about the conditions under which these structures are stable. The goal of this project is to obtain such criteria in a framework that extends earlier work to more general classes of waves and to broader settings that account for the effects of higher dimensions, lattice structure, external driving, and damping. Fundamental understanding of stability criteria for nonlinear waves is important in a number of different fields, including materials science, condensed matter physics, mechanical engineering, and biophysics. The insights provided by this work can be helpful in designing new devices for energy channeling, shock absorption, and vibration mitigation. An integral component of this collaborative project is teaching and training graduate students in the interdisciplinary research area of nonlinear wave phenomena in lattices. To this end, graduate students participate in the research.This project builds on the ongoing joint work of the principal investigators on stability of solitary traveling waves in one-dimensional lattices and its intimate connection to the stability of discrete breathers. This involves developing novel analytical stability criteria, as well as state-of-the-art numerical approaches necessary to extend this framework to broader settings, including two-dimensional and heterogeneous lattices, while considering more general classes of traveling waves. In particular, stability of waveforms with oscillatory tails, such as nanoptera in resonant granular chains and generalized kinks in dislocation models, is investigated. The project involves studying dynamic implications of instability and examining the effects of dissipation, long-range interactions, and external driving. The project's goals also include clarifying the relation of the linear spectra associated with a solitary traveling wave (and their variations at instability critical points) to the corresponding discrete breather Floquet spectra, as well as connecting the stability criteria for such waves to the established functional analytic framework for continuum systems. Graduate students participate in the research.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.