Nonlinear Waves in Lattices and Metamaterials

该项目研究空间离散系统中过渡前沿和其他非线性波的动力学。这些波在晶格和机械超材料中传输能量方面发挥着重要作用，这些超材料是利用其组件的不稳定性产生所需集体响应的工程结构。该项目旨在推进与波传播相关的能量转移现象的基本理解。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，这些特性使其能够应用于软机器人，变形表面，可重构器件和机械逻辑等。由于非线性跃迁波在物理和生物环境中的普遍存在，理解它们的性质以及它们存在和稳定的条件与机械工程，材料科学，凝聚态物理和生物物理等领域有关。该项目将为博士生提供研究培训机会。该项目将研究费米-帕斯塔-乌拉姆晶格及其各种扩展中的行波跃迁波，包括粘性耗散，双原子结构和长程相互作用。其目的是提供洞察的非线性，非凸性，耗散，异质性和非局部性对不同类型的过渡波的存在和稳定性的影响，以及在何种程度上这些影响可以被捕获准连续体近似。开放的基本问题，在过渡波的动力学将得到解决，包括内部的超音速扭结的过渡前的能量传递的分散机制和结构的动力学关系的亚音速扭结。该研究计划涉及分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，并构建分段线性最近邻相互作用问题的半解析解，这将有助于离散非线性系统的其他研究。构造的解决方案将作为基准测试准连续体近似的情况下，并通知调查的完全非线性问题。对不稳定性后果的数值探索将阐明解的结构，并可能揭示其他非线性waveforms.This奖项的分叉反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.