Stability of compact breathers in translationally-invariant nonlinear chains with flat dispersion bands

Nathan Perchikov, O. V. Gendelman

Research output: Contribution to journalArticlepeer-review

Abstract

The paper addresses compact oscillatory states (compact breathers) in translationally-invariant lattices with flat dispersion bands. The compact breathers appear in such systems even in the linear approximation. If the interactions are nonlinear, but comply with the flat-band symmetry, the compact breather solutions exist, but can lose their stability for certain parameter values. As benchmark nonlinear potentials, we use the β-FPU (Fermi-Pasta-Ulam) and vibro-impact models. Loss of stability is numerically observed to occur through either pitchfork or Hopf bifurcations. The loss of stability can occur through two qualitatively different mechanisms – through internal instability in the basic lattice elements, or through interaction of the compact breather with the linear passband of the lattice. The former scenario is more typical for high-amplitude breathers, and the latter – for low amplitudes. For the high-amplitude case, insights into the nature of compact-mode loss-of-stability are obtained by resorting to the limit of a piecewise-linear system, where interactions are represented by conservative impacts. This issue calls for detailed introspection into integrability of piecewise-linear (impacting) systems and their relation to the smooth system. An idea for a sensor based on the studied mechanisms is suggested.

Original languageEnglish
Article number109526
JournalChaos, Solitons and Fractals
Volume132
DOIs
StatePublished - Mar 2020

Keywords

  • Compact breathers
  • Cubic nonlinearity
  • Flat bands
  • Resonance
  • Translational invariance
  • Vibro-impact limit

All Science Journal Classification (ASJC) codes

  • Applied Mathematics
  • Statistical and Nonlinear Physics
  • General Physics and Astronomy
  • General Mathematics

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