Nonlinear waves in nonintegrable lattices

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

This award supports the research program of the Principal Investigator concerned with the behavior of waves in discrete (lattice) systems and potential applications related to this. Many physical and biological systems are both spatially discrete and nonlinear. Examples include crystal lattices, biopolymers, and nanoelectromechanical devices. The interplay of the discrete structure and nonlinearity often leads to the formation of spatially localized nonlinear waves that carry or trap energy. This project will advance fundamental understanding of properties and stability of nonlinear waves in discrete structures, which is important in a number of different fields, including materials science, condensed matter physics, mechanical engineering, and biophysics. It will also provide insight into the dynamics of nonlinear waves in two new types of tunable granular metamaterials and develop novel approaches to channeling and trapping of acoustic energy, with potential applications to energy harvesting, shock absorption, and vibration mitigation. An integral component of the project is teaching and training graduate students in the interdisciplinary research area of nonlinear lattice dynamics. The project will develop a new analytical approach to illuminate the essential role of discreteness in the dynamics of solitary waves in nonintegrable Hamiltonian lattices. It will clarify how this dynamics is affected by lattice dimension, nonlocality, nonconvexity, anisotropy, and heterogeneity, which often play an important role in real physical and biological discrete systems. The work will be accomplished in part by constructing semi-analytical solutions for one- and two-dimensional lattices with piecewise linear interactions. This will provide insight into fully nonlinear lattice problems, which will be studied through a combination of asymptotic and numerical methods. A particularly interesting class of nonintegrable lattices corresponding to locally resonant granular crystals, a new type of granular material that has additional internal degrees of freedom, will be considered. Through a combination of analytical, numerical, and experimental work, the PI and her collaborators will investigate the dynamics of solitary waves, breathers, and other nonlinear waves in these systems, focusing on the phenomena of energy trapping, wave redirection, and attenuation, which are of both fundamental and practical interest.