Large-amplitude nonlinear normal modes of the discrete sine lattices

Valeri V. Smirnov and Leonid I. Manevitch
Phys. Rev. E 95, 022212 – Published 17 February 2017

Abstract

We present an analytical description of the large-amplitude stationary oscillations of the finite discrete system of harmonically coupled pendulums without any restrictions on their amplitudes (excluding a vicinity of π). Although this model has numerous applications in different fields of physics, it was studied earlier in the infinite limit only. The discrete chain with a finite length can be considered as a well analytical analog of the coarse-grain models of flexible polymers in the molecular dynamics simulations. The developed approach allows to find the dispersion relations for arbitrary amplitudes of the nonlinear normal modes. We emphasize that the long-wavelength approximation, which is described by well-known sine-Gordon equation, leads to an inadequate zone structure for the amplitudes of about π/2 even if the chain is long enough. An extremely complex zone structure at the large amplitudes corresponds to multiple resonances between nonlinear normal modes even with strongly different wave numbers. Due to the complexity of the dispersion relations the modes with shorter wavelengths may have smaller frequencies. The stability of the nonlinear normal modes under condition of the resonant interaction are discussed. It is shown that this interaction of the modes in the vicinity of the long wavelength edge of the spectrum leads to the localization of the oscillations. The thresholds of instability and localization are determined explicitly. The numerical simulation of the dynamics of a finite-length chain is in a good agreement with obtained analytical predictions.

  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
  • Figure
1 More
  • Received 30 March 2016
  • Revised 25 November 2016

DOI:https://doi.org/10.1103/PhysRevE.95.022212

©2017 American Physical Society

Physics Subject Headings (PhySH)

  1. Research Areas
  1. Physical Systems
Nonlinear Dynamics

Authors & Affiliations

Valeri V. Smirnov* and Leonid I. Manevitch

  • Institute of Chemical Physics, RAS, 4 Kosygin Street, Moscow 119991, Russia

  • *vvs@polymer.chph.ras.ru

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand
Issue

Vol. 95, Iss. 2 — February 2017

Reuse & Permissions
Access Options
Author publication services for translation and copyediting assistance advertisement

Authorization Required


×
×

Images

×

Sign up to receive regular email alerts from Physical Review E

Log In

Cancel
×

Search


Article Lookup

Paste a citation or DOI

Enter a citation
×