Finite size effects on instabilities of discrete breathers

Abstract

When the two arcs of the continuous phonon spectrum of the Floquet matrix of a discrete breather overlap on the unit circle, the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable. More precisely, when the model parameters vary, the breather in the finite system exhibits a large number of collisions between the Floquet eigenvalues belonging to the phonon spectrum. These collisions correspond to complex cascades of instability thresholds followed near after by re-entrant stability thresholds. We interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action. Then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges, but simultaneously the maximum amplitude of the instabilities vanishes, so that the breather in the infinite system recovers its linear stability (as long as all its other localized modes remain stable). This is the situation which is required in Cretegny et al. [Physica D 119 (1998) 73–87] for having inelastic phonon scattering with two channels. We also analyze the size effects when a Floquet eigenvalue associated with a localized mode collides with the Floquet continuous phonon spectrum with different Krein signature. In contrast to the previous case, the infinite system is unstable after the collision.