Multibreathers in Klein-Gordon chains with interactions beyond nearest neighbors

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 {\it crucial} modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in- or anti-phase. Configurations with nontrivial phase profiles, arise, as well as spontaneous symmetry breaking (pitchfork) bifurcations, when these states emerge from (or collide with) the ones with standard (0 or $\pi$) phase difference profiles. Similar bifurcations, both of the supercritical and of the subcritical type 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

On the stability of multibreathers in Klein-Gordon chains

In the present paper, a theorem, which determines the linear stability of multibreathers in Klein-Gordon chains, is proven. Specifically, it is shown that for soft nonlinearities, and positive 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 theorem can be applied in $n$-site breathers, where $n$ is any finite number and provides an $\cal{O}(\sqrt{\epsilon})$ estimation of the characteristic exponents of the solution. 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 $\epsilon$

Existence of multi-site intrinsic localized modes in one-dimensional Debye crystals

The existence of highly localized multi-site oscillatory structures (discrete multibreathers) in a nonlinear Klein-Gordon chain which is characterized by an inverse dispersion law is proven and their linear stability is investigated. The results are applied in the description of vertical (transverse, off-plane) dust grain motion in dusty plasma crystals, by taking into account the lattice discreteness and the sheath electric and/or magnetic field nonlinearity. Explicit values from experimental plasma discharge experiments are considered. The possibility for the occurrence of multibreathers associated with vertical charged dust grain motion in strongly-coupled dusty plasmas (dust crystals) is thus established. From a fundamental point of view, this study aims at providing a first rigorous investigation of the existence of intrinsic localized modes in Debye crystals and/or dusty plasma crystals and, in fact, suggesting those lattices as model systems for the study of fundamental crystal properties.Comment: 12 pages, 8 figures, revtex forma

Discrete breathers in Φ4 and related models

In this Chapter, we touch upon the wide topic of discrete breather (DB) formation with a special emphasis on the prototypical system of interest, namely the 4 model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the DBs. Our main emphasis is on the existence, and especially on the stability features of such solutions.We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted. While this Chapter by no means aspires to be comprehensive, we hope that it provides some recent developments (a large fraction of which is not included in time-honored DB reviews) and associated future possibilities.AEI/FEDER, (UE) MAT2016- 79866-

Discrete breathers in ϕ4\phi^4 and related models

We touch upon the wide topic of discrete breather formation with a special emphasis on the the $\phi^4$ model. We start by introducing the model and discussing some of the application areas/motivational aspects of exploring time periodic, spatially localized structures, such as the discrete breathers. Our main emphasis is on the existence, and especially on the stability features of such solutions. We explore their spectral stability numerically, as well as in special limits (such as the vicinity of the so-called anti-continuum limit of vanishing coupling) analytically. We also provide and explore a simple, yet powerful stability criterion involving the sign of the derivative of the energy vs. frequency dependence of such solutions. We then turn our attention to nonlinear stability, bringing forth the importance of a topological notion, namely the Krein signature. Furthermore, we briefly touch upon linearly and nonlinearly unstable dynamics of such states. Some special aspects/extensions of such structures are only touched upon, including moving breathers and dissipative variations of the model and some possibilities for future work are highlighted

On the nonexistence of degenerate phase-shift multibreathers in Klein-Gordon models with interactions beyond nearest neighbors

In this work, we study the existence of, low amplitude, phase-shift multibreathers for small values of the linear coupling in KleinGordon chains with interactions beyond the classical nearest-neighbor (NN) ones. In the proper parameter regimes, the considered lattices bear connections to models beyond one spatial dimension, namely the so-called zigzag lattice, as well as the two-dimensional square lattice or coupled chains. We examine initially the necessary persistence conditions of the system derived by the so-called Effective Hamiltonian Method, in order to seek for unperturbed solutions whose continuation is feasible. Although this approach provides useful insights, in the presence of degeneracy, it does not allow us to determine if they constitute true solutions of our system. In order to overcome this obstacle, we follow a different route. By means of a Lyapunov-Schmidt decomposition, we are able to establish that the bifurcation equation for our models can be considered, in the small energy and small coupling regime, as a perturbation of a corresponding, beyond nearest-neighbor, discrete nonlinear Schr\ua8odinger equation. There, nonexistence results of degenerate phase-shift discrete solitons can be demonstrated by an additional Lyapunov-Schmidt decomposition, and translated to our original problem on the Klein-Gordon system. In this way, among other results, we can prove nonexistence of four-sites vortex-like waveforms in the zigzag Klein-Gordon model. Finally, briefly considering a one-dimensional model bearing similarities to the square lattice, we conclude that the above strategy is not efficient for the proof of the existence or nonexistence of vortices due to the higher degeneracy of this configuration

Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations

We prove the most general theorem about spectral stability of multi-site breathers in the discrete Klein-Gordon equation with a small coupling constant. In the anti-continuum limit, multi-site breathers represent excited oscillations at different sites of the lattice separated by a number of "holes" (sites at rest). The theorem describes how the stability or instability of a multi-site breather depends on the phase difference and distance between the excited oscillators. Previously, only multi-site breathers with adjacent excited sites were considered within the first-order perturbation theory. We show that the stability of multi-site breathers with one-site holes change for large-amplitude oscillations in soft nonlinear potentials. We also discover and study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure