Collaborative Research: Stability of Nonlinear Wave Structures in Lattices

University Of Massachusetts Amherst, Amherst MA

Investigators

Abstract

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.