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

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 $\beta$-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.