Abstract. In the present paper, a theorem, which determines the linear stability of multibreathers excited over adjacent coupled oscillators in Klein-Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive nearestneighbor inter-site coupling, only structures with adjacent sites excited out-of-phase may be stable, while only in-phase ones may be stable for negative coupling. The situation is reversed for hard nonlinearities. This method can be applied to n-site breathers, where n is any finite number and provides a detailed count of the number of real and imaginary characteristic exponents of the breather, based on its configuration. In addition, an O( √ ε) estimation of these exponents can be extracted through this procedure. To complement the analysis, we perform numerical simulations and establish that the results are in excellent agreement with the theoretical predictions, at least for small values of the coupling constant ε.

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 φ 4 model.

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In this work we use standard Hamiltonian-system techniques in order to study the dynamics of three vortices with alternating charges in a confined Bose-Einstein condensate. In addition to being motivated by recent experiments, this system offers a natural vehicle for the exploration of the transition of the vortex dynamics from ordered to progressively chaotic behavior. In particular, it possesses two integrals of motion, the energy (which is expressed through the Hamiltonian H) and the angular momentum L of the system. By using the integral of the angular momentum, we reduce the system to a 2-degrees-of-freedom one with L as a parameter and reveal the topology of the phase space through the method of Poincaré surfaces of section. We categorize the various motions that appear in the different regions of the sections and we study the major bifurcations that occur to the families of periodic motions of the system. Finally, we correspond the orbits on the surfaces of section to the real space motion of the vortices in the plane.