Collaborative Research: Stability of Nonlinear Wave Structures in Lattices

这一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、纳米机电设备到生物聚合物，空间离散的结构和非线性相互作用，经常导致形成非线性波形，当能量在系统中传播时，这些波形要么相干地传输，要么释放。特别是，最近关于颗粒材料的实验和理论工作促进了人们对各种此类结构的存在条件和性质的理解，包括行波脉冲和冲击波。然而，关于这些结构在什么条件下是稳定的，人们所知的要少得多。这个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浪类别和更广泛的设置，这些设置考虑了更高维度、格子结构、外部驱动和阻尼的影响。基本了解非线性波的稳定性准则在材料科学、凝聚态物理、机械工程和生物物理学等许多不同领域中都很重要。这项工作提供的见解可以帮助设计新的能量通道、减震和减振装置。这个合作项目的一个组成部分是在格子中的非线性波动现象的跨学科研究领域教授和培训研究生。为此，研究生参与了这项研究。这个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稳定性的密切联系的共同工作的基础上。这包括开发新的分析稳定性标准，以及必要的最先进的数值方法，以将该框架扩展到更广泛的环境，包括二维和非均匀晶格，同时考虑更一般的行波类别。特别地，研究了具有振荡尾部的波形的稳定性，如共振颗粒链中的纳米晶和位错模型中的广义扭结。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Floquet谱的关系，以及将此类波的稳定性标准与已建立的连续统系统的功能分析框架联系起来。研究生参与研究。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.