Nonlinear Instabilities of Multi‐Site Breathers in Klein–Gordon Lattices

Abstract

In the present work, we explore the possibility of excited breather states in a nonlinear Klein-Gordon lattice to become nonlinearly unstable, even if they are found to be spectrally stable. The mechanism for this fundamentally nonlinear instability is through the resonance with the wave continuum of a multiple of an internal mode eigenfrequency in the linearization of excited breather states. For the nonlinear instability, the internal mode must have its Krein signature opposite to that of the wave continuum. This mechanism is not only theoretically proposed, but also numerically corroborated through two concrete examples of the KleinGordon lattice with a soft (Morse) and a hard (φ 4 ) potential. Compared to the case of the nonlinear Schrödinger lattice, the Krein signature of the internal mode relative to that of the wave continuum may change depending on the period of the excited breather state. For the periods for which the Krein signatures of the internal mode and the wave continuum coincide, excited breather states are observed to be nonlinearly stable.

Approximation of small-amplitude weakly coupled oscillators by discrete nonlinear Schrödinger equations

“…compared with the previous estimate (13). As a result, the justification analysis developed in the proof of Theorems 1 and 2 holds verbatim and results in the following theorems.…”

Energy Criterion for the Spectral Stability of Discrete Breathers

“…Spectral stability of excited (multi-site) breathers was classified near the AC limit in the work of [22][23][24], depending on the phase difference in the nonlinear oscillations between different sites of the lattice. More recently, nonlinear instability of spectrally stable two-site breathers was shown in [25]. Nevertheless, an overarching criterion of breather stability tantamount to the VK criterion remains un-known up to now.…”

Discrete Breathers in $$\phi ^4$$ and Related Models

“…Finally, the right panels of the figure consider a linearly stable but nonlinearly unstable multibreather with σ = {1, 1} displaying a similar behaviour. The nonlinearly unstable dynamics fulfills the conditions of the theorem of [73] as the frequency of the internal mode is Ω = 0.2073 and the spectral bands expands in [0.32, 0.5]. As a result, the second harmonic of the internal mode lies in the phonon arc, ultimately (at very long times) slowly feeding the nonlinear growth and eventually leading to the destruction of the two-site configuration.…”

“…Having in mind the stability properties of multibreathers near the AC limit, the only stable solutions among them are those whose codes σ are in phase if the on-site potential is hard and in anti-phase if the potential is soft. As demonstrated in [73], the internal modes that detaches from θ = 0 have, in the upper half-circle (i.e. for θ ∈ [0, π], Krein signature κ = 1 if the potential is soft and κ = −1 if it is hard.…”

“…Based in such findings, in [73] we explored the nonlinear instability of linearly stable multibreathers. We found that nonlinear instability is possible when an integer multiple of the frequency of an internal (i.e.…”