The monoatomic FPU system as a limit of a diatomic FPU system

Abstract

We consider a diatomic infinite Fermi-Pasta-Ulam (FPU) system with light and heavy particles. For a small mass ratio, we prove error estimates for the approximation of the dynamics of this system by the dynamics of the monoatomic FPU system. The light particles are squeezed by the heavy particles at the mean value of their displacements. The error estimates are derived by means of the energy method and hold for sufficiently long times, for which the dynamics of the monoatomic FPU system is observed. The approximation result is restricted to sufficiently small displacements of the heavy particles relatively to each other.

Micropterons, Nanopterons and Solitary Wave Solutions to the Diatomic Fermi-Pasta-Ulam-Tsingou Problem

“…Physically, this means that the mass-less particles are fixed halfway between the heavier ones, corresponding with the intuition developed in [38,56]. Upon setting…”

Mass‐in‐mass lattices with small internal resonators

“…This paper is, in part, an attempt to address similar issues for the Cauchy problem. We also mention the article by Pelinovsky and Schneider 9 . In that paper, the authors treat a diatomic FPUT lattice in the limit that the mass ratio tends to zero.…”

“…Remark As we mentioned in the introduction, the article 9 treats the monatomic limit of a diatomic FPUT lattice in the case of small mass ratio. Their mass ratio is named and is most comparable to our internal mass .…”

Mass and spring dimer Fermi–Pasta–Ulam–Tsingou nanopterons with exponentially small, nonvanishing ripples

“…We mention just a handful of related results here and discuss others in the context of future problems in Section 5. Pelinovsky‐Schneider 25 considered the dimer small mass limit as an initial value problem in $$\ell ^2(\mathbb {Z})$$‐type sequence spaces and found that if the initial data are close to a solution of the limiting monatomic lattice and the mass ratio is sufficiently small, then the dimer solution remains close to that monatomic solution. McGinnis and Wright 26 moved well beyond the polyatomic regime to find that the classical wave equation is a good approximation for linear FPUT lattices with “random” material data, that is, their potentials are $$\mathcal {V}_j^{\prime }(r) = \kappa _jr$$, where $$\kappa _j$$ and the masses $$m_j$$ are all random variables.…”