Nonlinear instabilities of multi-site breathers in Klein-Gordon lattices

TL;DR

This paper investigates how excited breather states in nonlinear Klein-Gordon lattices can become nonlinearly unstable through resonance mechanisms, even when spectrally stable, with numerical validation in specific potential models.

Contribution

It introduces a new nonlinear instability mechanism based on Krein signature resonance and demonstrates it with numerical examples in Klein-Gordon lattices.

Findings

Nonlinear instability occurs via resonance with wave continuum.

Krein signature mismatch leads to instability.

Numerical examples confirm theoretical predictions.

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 Klein--Gordon lattice with a soft (Morse) and a hard ($\phi^4$) potential. Compared to the case of the nonlinear Schr{\"o}dinger 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.