Collaborative Research: Stability of Nonlinear Wave Structures in Lattices

该合作研究项目的主要研究人员研究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设备到生物聚合物，空间离散结构和非线性的相互作用通常导致非线性波形的形成，当它们通过系统传播时，非线性波形相干地传输或释放能量。 特别是，最近的实验和理论工作的粒状材料的存在条件和性能的各种结构，包括行波脉冲和冲击波的理解。 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和更广泛的设置，以考虑更高维度，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。 这项工作所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教学和培训研究生在非线性波现象的跨学科研究领域的晶格。 为此目的，研究生参与了研究。该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稳定性及其与离散呼吸子稳定性的密切联系的联合工作基础上。 这涉及到开发新的分析稳定性标准，以及国家的最先进的数值方法，必要的扩展这一框架，以更广泛的设置，包括二维和异质晶格，同时考虑更一般的类行波。 特别是，振荡的尾巴，如纳米翅共振颗粒链和广义扭结位错模型的波形的稳定性进行了研究。 该项目涉及研究不稳定性的动态影响，并检查耗散，长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连续系统的既定功能分析框架相连接。 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，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.