Energy criterion for the spectral stability of discrete breathers

TL;DR

This paper introduces an energy-based criterion for predicting the spectral stability of discrete breathers in nonlinear lattices, linking stability to the monotonicity of energy with respect to frequency.

Contribution

It proposes a novel stability criterion for discrete breathers analogous to the Vakhitov-Kolokolov criterion used for solitary waves.

Findings

Breathers with increasing energy as a function of frequency are unstable in soft potentials.

Breathers with decreasing energy as a function of frequency are unstable in hard potentials.

Numerical results support the proposed energy criterion for stability prediction.

Abstract

Discrete breathers are ubiquitous structures in nonlinear anharmonic models ranging from the prototypical example of the Fermi-Pasta-Ulam model to Klein-Gordon nonlinear lattices, among many others. We propose a general criterion for the emergence of instabilities of discrete breathers analogous to the well-established Vakhitov-Kolokolov criterion for solitary waves. The criterion involves the change of monotonicity of the discrete breather's energy as a function of the breather frequency. Our analysis suggests and numerical results corroborate that breathers with increasing (decreasing) energy-frequency dependence are generically unstable in soft (hard) nonlinear potentials.