Mass-in-Mass Lattices with Small Internal Resonators

TL;DR

This paper studies mass-in-mass lattices with tiny internal resonators, showing they closely follow standard FPUT systems over long times and providing detailed error estimates for this approximation.

Contribution

It demonstrates that small internal resonators cause the MiM lattice to shadow FPUT dynamics for extended periods, with precise error bounds derived.

Findings

MiM lattice solutions shadow FPUT solutions over long times

Error estimates for the approximation are higher than expected

Classical oscillatory integral estimates are used to analyze the error

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.