Nonlinear Waves in Lattices and Metamaterials

该项目研究空间离散系统中过渡前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发挥着重要作用，这些超材料是利用其组件的不稳定性来产生所需的集体响应的工程结构。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，这些超材料能够应用于软机器人、变形表面、可重构设备和机械逻辑等领域。由于非线性跃迁波在物理和生物环境中的普遍存在，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等领域具有重要意义。该项目将为博士生提供研究培训机会。该项目将研究Fermi-Pasta-Ulam晶格中的行波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。其目的是深入了解非线性、非凸性、耗散、非均质性和非局域性对不同类型过渡波的存在和稳定性的影响，以及这些影响中的一些可以被拟连续近似捕捉的程度。文中将讨论转变波动力学中尚未解决的基本问题，包括超音速扭结的能量传递的色散机制和亚音速扭结的动力学关系的结构。该研究项目包括发展解析和数值方法，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近邻相互作用问题的半解析解，这将对离散非线性系统的其他研究有用。构造的解将作为测试拟连续近似的基准案例，并为完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澄清解决方案的结构，并可能揭示其他非线性波形的分叉。该奖项反映了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 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.