Nonlinear Waves in Lattices and Metamaterials

该项目研究了空间离散系统中过渡锋和其他非线性波的动力学。这些波在晶格和机械超材料中的能量传输中起着重要作用，这些工程结构利用其组件的不稳定性来产生期望的集体响应。该项目旨在促进对与波传播相关的能量传递现象的基本理解。这些信息对于设计具有所需特性的新型机械超材料非常重要，这些材料可以应用于软机器人、变形表面、可重构设备和机械逻辑等领域。由于非线性过渡波在物理和生物环境中无处不在，了解它们的性质，以及它们存在和稳定的条件，在机械工程、材料科学、凝聚态物理和生物物理学等领域都是相关的。本项目将为博士生提供科研培训机会。本项目将研究Fermi-Pasta-Ulam晶格中的行跃迁波及其各种扩展，包括粘性耗散、双原子结构和远程相互作用。目的是深入了解非线性、非凸性、耗散、非均匀性和非局域性对不同类型过渡波的存在和稳定性的影响，以及准连续统近似可以捕捉其中一些影响的程度。在过渡波动力学开放的基本问题将被解决，包括能量转移的色散机制内的超音速扭结和亚音速扭结的动力学关系的结构。该研究计划涉及解析和数值方法的发展，例如将行波表示为非线性映射的不动点，以及构造具有分段线性最近邻相互作用的问题的半解析解，这将在其他离散非线性系统的研究中有用。所构造的解将作为测试准连续统近似的基准案例，并为全非线性问题的研究提供信息。对不稳定后果的数值探索将阐明解的结构，并可能揭示其他非线性波形的分岔。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 project investigates the dynamics of transition fronts and other nonlinear waves in spatially discrete systems. These waves play a major role in transporting energy in lattices and mechanical metamaterials, engineered structures that exploit instabilities of their components to yield a desired collective response. The project aims to advance the fundamental understanding of the energy transfer phenomena associated with the wave propagation. This information is important for designing novel mechanical metamaterials with desired characteristics that enable applications in soft robotics, morphing surfaces, reconfigurable devices, and mechanical logic, among others. Due to the ubiquity of nonlinear transition waves in physical and biological settings, understanding their properties, as well as the conditions for their existence and stability, is relevant in fields such as mechanical engineering, materials science, condensed matter physics, and biophysics. The project will provide research training opportunities for doctoral students.This project will study traveling transition waves in a Fermi-Pasta-Ulam lattice and its various extensions that include viscous dissipation, diatomic structure, and long-range interactions. The intent is to provide insights into the effects of nonlinearity, nonconvexity, dissipation, heterogeneity, and nonlocality on the existence and stability of different types of transition waves and the extent to which some of these effects can be captured by quasicontinuum approximations. Open fundamental questions in the dynamics of transition waves will be addressed, including the dispersive mechanism of energy transfer inside the transition front of a supersonic kink and the structure of the kinetic relations for subsonic kinks. The research program involves the development of analytical and numerical approaches, such as representing traveling waves as fixed points of a nonlinear map and constructing semi-analytical solutions of the problems with piecewise linear nearest-neighbor interactions, that will be useful in other studies of discrete nonlinear systems. The constructed solutions will serve as benchmark cases for testing quasicontinuum approximations and inform the investigation of the fully nonlinear problems. Numerical explorations of the consequences of instability will clarify the solution structure and possibly reveal bifurcations of other nonlinear waveforms.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.