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.
Highlights
1.1 The Mass-in-Mass LatticeConsider an infinite chain of particles, which we call “beads,” each normalized to have mass 1, arranged on a horizontal line and connected by identical springs whose potential is V ∈ C 1(R)
A finite, one-dimensional chain of particles is constructed; the chain is struck at one end by another particle, and the resulting motion of the chain is measured [6]. Common to these experiments is an interest in the propagation of traveling waves through the granular crystal, which are waves whose form is given by a steadily translating profile [38,39]
Our intention is to perturb from a monatomic solitary wave, in an appropriate sense, and construct traveling wave solutions for the MiM lattice when μ is small
Summary
Consider an infinite chain of particles, which we call “beads,” each normalized to have mass 1, arranged on a horizontal line and connected by identical springs whose potential is V ∈ C 1(R). Springs connecting the beads to the resonators are all identical and exert the force κr when stretched a distance r. This construct is called a mass-in-mass (MiM) lattice. A finite, one-dimensional chain of particles is constructed; the chain is struck at one end by another particle, and the resulting motion of the chain is measured [6] Common to these experiments is an interest in the propagation of traveling waves through the granular crystal, which are waves whose form is given by a steadily translating profile [38,39]. As representative of the diversity of applications of MiM lattices, we mention that they arise in constructing sensors for bone elasticity [52] and ultrasonic scans [42], determining of the setting time for cement [40], and modeling of switches and logic gates [33]
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