Multibreather and vortex breather stability in Klein--Gordon lattices: Equivalence between two different approaches

TL;DR

This paper compares two analytical methods for assessing the stability of multibreather and vortex breather solutions in Klein-Gordon lattices, demonstrating their equivalence and applying them to various lattice models.

Contribution

It introduces and proves the equivalence of the Aubry band and MacKay effective Hamiltonian methods for stability analysis of multibreathers.

Findings

The two methods yield identical stability predictions.

Applications to 1D and 2D lattice models validate the approaches.

Results enhance understanding of breather stability in nonlinear lattices.

Abstract

In this work, we revisit the question of stability of multibreather configurations, i.e., discrete breathers with multiple excited sites at the anti-continuum limit of uncoupled oscillators. We present two methods that yield quantitative predictions about the Floquet multipliers of the linear stability analysis around such exponentially localized in space, time-periodic orbits, based on the Aubry band method and the MacKay effective Hamiltonian method and prove that their conclusions are equivalent. Subsequently, we showcase the usefulness of the methods by a series of case examples including one-dimensional multi-breathers, and two-dimensional vortex breathers in the case of a lattice of linearly coupled oscillators with the Morse potential and in that of the discrete $\phi^4$ model.