Energy Criterion for the Spectral Stability of Discrete Breathers

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.

Figure 1

Breathers in a 2D KG lattice with a hard quartic potential in the case of $C=0.5$ (left panels) and a Morse potential with $C=0.2$ (right panels). The top panels show the profile of two unstable breathers with a portion of the unit circle shown in the inset, corresponding to $C=0.5$, $\omega =2.3$ (left) and $C=0.2$, $\omega =0.992$ (right). Central panels shows the energy-frequency dependence, whereas the bottom panels display the Floquet multipliers with $|\mu |>1$ (i.e., associated with instability) versus $\omega$.

Figure 2

Breathers in a monoatomic FPU chain with $\alpha =-1$, $\beta =1$. Left (right) panels correspond to the bond-centered (site-centered) mode. The top panels show the breather profiles, in the strain variable, for $\omega =2.1$. The middle panel shows the energy-frequency dependence, whereas the bottom panel displays modulus of the Floquet multipliers with $|\mu |>1$ versus $\omega$.

Figure 3

Gap breathers in a diatomic FPU chain for a hard potential with $\alpha =-1$, $\beta =1$, $\epsilon =0.8$ (left) and a soft potential with $\alpha =0$, $\beta =-1$, and $\epsilon =0.7$ (right). The top panels show the breather profiles, in the strain variable, for $\omega =1.7$ (left) and $\omega =1.4$ (right). Blue (red) dots correspond to the more (less) massive particles. The middle panel shows the energy-frequency dependence, whereas the bottom panel displays the modulus of the Floquet multipliers with $|\mu |>1$ versus $\omega$.