Small mass nanopteron traveling waves in mass-in-mass lattices with cubic FPUT potential

TL;DR

This paper proves the existence of small mass nanopteron traveling waves in a mass-in-mass lattice with cubic FPUT potential, revealing complex wave structures combining localized and periodic components.

Contribution

It introduces a rigorous proof of nanopteron wave existence in a mass-in-mass lattice with small resonator mass, extending previous analytical techniques to this nonlinear lattice model.

Findings

Existence of nanopteron traveling waves in the small mass limit.

Wave profiles combine localized core and periodic oscillation.

Applicable functional analytic techniques from water wave theory.

Abstract

The mass-in-mass (MiM) lattice consists of an infinite chain of identical beads that are both nonlinearly coupled to their nearest neighbors and linearly coupled to a distinct resonator particle; it serves as a prototypical model of wave propagation in granular crystals and metamaterials. We study traveling waves in an MiM lattice whose bead interaction is governed by the cubic Fermi-Pasta-Ulam-Tsingou potential and whose resonator mass is small compared to the bead mass. Excluding a countable number of "antiresonance" resonator masses accumulating at 0, we prove the existence of nanopteron traveling waves in this small mass limit. The profiles of these waves consist of the superposition of an exponentially localized core and a small amplitude periodic oscillation that itself is a traveling wave profile for the lattice. Our arguments use functional analytic techniques originally developed by Beale for a capillary-gravity water wave problem and recently employed in a number of related nanopteron constructions in diatomic Fermi-Pasta-Ulam-Tsingou lattices.