bibkey: rodrigueskevrekidisdobson2019nbreak
authors: A. S. Rodrigues; P. G. Kevrekidis; M. Dobson
year: 2019
title: "N-break states in a chain of nonlinear oscillators"
doi: 10.1103/PhysRevE.99.022201
url: "https://arxiv.org/abs/1802.03477v2"
claim: "For a clamped, pre-stretched chain of oscillators with the Lennard-Jones interaction, solutions with k broken bonds are computed for k up to 5, and the abstract conjectures that k-break solutions possess at least k-1 and at most k pairs of unstable eigenvalues."
strata_touched:
  - D5/S3/StatisticalMechanics/LennardJonesChainBreakInstabilityCount
license: citation-only
triage: anchor

Rodrigues, Kevrekidis and Dobson, N-break states in a chain of nonlinear oscillators

A. S. Rodrigues, P. G. Kevrekidis and M. Dobson, Phys. Rev. E 99(2), 022201 (2019); arXiv:1802.03477v2. Quotations are from the v2 TeX source.

The model has $M$ free oscillators with displacements $u_0,\dots ,u_{M+1}$, clamped by $u_0=u_{M+1}=0$ and pre-stretched to separation $d$, with potential energy $V(\mathbf{u})={\sum }_{n=1}^{M+1}[\varphi (d+u_n-u_{n-1})-\varphi (d)]$ and equations of motion ${\ddot{u}}_n={\varphi }^{\prime }(d+u_{n+1}-u_n)-{\varphi }^{\prime }(d+u_n-u_{n-1})$ for $n=1,\dots ,M$. The interaction is $\varphi (r)=r^{-12}-2r^{-6}$, with inflection point $r_i=(13/7{)}^{1/6}$. Linearizing about a steady state gives ${\lambda }^2\mathbf{\delta }=J({\mathbf{u}}_0)\mathbf{\delta }$ with $J$ the Jacobian of the force field; positive ${\lambda }^2$ is a dynamical instability. A bond of length $r<r_i$ is elastic and one of length $r>r_i$ is broken. The abstract states:

Each of these higher ``excited state’’ solutions possesses an additional unstable pair of eigenvalues. We thus conjecture that $k$ break solutions will possess at least $k-1$ (and at most $k$) pairs of unstable eigenvalues.

Verified locator

Source: https://arxiv.org/abs/1802.03477v2

• DOI: https://doi.org/10.1103/PhysRevE.99.022201 (Physical Review E 99(2), 022201; title, authors, journal, volume, issue and article number from Crossref).
• URL: https://arxiv.org/abs/1802.03477v2. The e-print member oscillatorsNN_resubmit.tex has 1019 lines and md5 f0e3a2d64045c090b53b217d75e86a05. Locations:
  • the abstract and the conjecture, l. 44–60 (the conjecture at l. 55–59);
  • the clamped chain, its potential energy and equations of motion, l. 160–181; the Lennard-Jones potential and the inflection point, l. 184–200;
  • the linearization and the Jacobian, l. 225–250; elastic and broken bonds, l. 280–283.