Mass‐in‐mass lattices with small internal resonators

Abstract

We consider the mass-in-mass (MiM) lattice when the internal resonators are very small. When there are no internal resonators the lattice reduces to a standard Fermi-Pasta-Ulam-Tsingou (FPUT) system. We show that the solution of the MiM system, with suitable initial data, shadows the FPUT system for long periods of time. Using some classical oscillatory integral estimates we can conclude that the error of the approximation is (in some settings) higher than one may expect.

Korteweg–de Vries waves in peridynamical media

“…Moreover, Refs. 40 and 41 concern the KdV limit in FPUT chains with periodically varying parameters and mass‐in‐mass lattices, respectively.…”

A simple model of radiating solitary waves

“…The observed attenuation is so slow that one of the major tools for analyzing the dynamics of solitary wave-like solutions, namely approximations of the problem with the Korteweg-de Vries (KdV) (or similar) equations [12,30,15,6,29], is incapable of capturing the phenomena. Such approximations are valid over very long but nevertheless finite time intervals; the erosion is so subtle during the period of good approximation that it falls within the natural error bounds.…”