50 research outputs found
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 ) 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 -site breathers, where is
any finite number and provides an 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
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
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