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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We consider a one-dimensional discrete nonlinear Schrödinger (dNLS) model featuring interactions beyond nearest neighbors. We are interested in the existence (or nonexistence) of phase-shift discrete solitons, which correspond to four-sites vortex solutions in the standard two-dimensional dNLS model (square lattice), of which this is a simpler variant. Due to the specific choice of lengths of the inter-site interactions, the vortex configurations considered present a degeneracy which causes the standard continuation techniques to be non-applicable.In the present one-dimensional case, the existence of a conserved quantity for the soliton profile (the so-called density current), together with a perturbative construction, leads to the nonexistence of any phase-shift discrete soliton which is at least C 2 with respect to the small coupling ǫ, in the limit of vanishing ǫ. If we assume the solution to be only C 0 in the same limit of ǫ, nonexistence is instead proved by studying the bifurcation equation of a Lyapunov-Schmidt reduction, expanded to suitably high orders. Specifically, we produce a nonexistence criterion whose efficiency we reveal in the cases of partial and full degeneracy of approximate solutions obtained via a leading order expansion.

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.

We consider a prototypical dynamical lattice model, namely, the discrete nonlinear Schrödinger equation on nonsquare lattice geometries. We present a systematic classification of the solutions that arise in principal six-lattice-site and three-lattice-site contours in the form of both discrete multipole solitons and discrete vortices. Additionally to identifying the possible states, we analytically track their linear stability both qualitatively and quantitatively. We find that among the six-site configurations, the "hexapole" of alternating phases (0-pi) , as well as the vortex of topological charge S=2 have intervals of stability; among three-site states, only the vortex of topological charge S=1 may be stable in the case of focusing nonlinearity. These conclusions are confirmed both for hexagonal and for honeycomb lattices by means of detailed numerical bifurcation analysis of the stationary states from the anticontinuum limit, and by direct simulations to monitor the dynamical instabilities, when the latter arise. The dynamics reveal a wealth of nonlinear behavior resulting not only in single-site solitary wave forms, but also in robust multisite breathing structures.

We study the existence and stability of multibreathers in Klein-Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in-or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard (0 or π) phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either next-nearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined. Our analytical predictions are found to be in very good agreement with numerical results.

We prove the existence of monoparametric families of multibreathers in chains of Hamiltonian oscillators of one degree of freedom, with the total energy of the chain as a parameter. At the same time we evaluate the neighborhood of the initial conditions for these solutions, as well as their stability. For the proof we use an idea originally proposed by Poincaré. We apply our results to calculate families of multibreathers in a chain consisting of coupled Morse oscillators.