Nonintegrable Schrodinger discrete breathers

In an extensive numerical investigation of nonintegrable translational motion of discrete breathers in nonlinear Schr?dinger lattices, we have used a regularized Newton algorithm to continue these solutions from the limit of the integrable Ablowitz-Ladik lattice. These solutions are shown to be a superposition of a localized moving core and an excited extended state (background) to which the localized moving pulse is spatially asymptotic. The background is a linear combination of small amplitude nonlinear resonant plane waves and it plays an essential role in the energy balance governing the translational motion of the localized core. Perturbative collective variable theory predictions are critically analyzed in the light of the numerical results.     

Linearity stabilizes discrete breathers

The study of the dynamics of 1D chains with both harmonic and nonlinear interactions, as in the Fermi–Pasta–Ulam (FPU) and related problems, has played a central role in efforts to identify the broad consequences of nonlinearity in these systems. Here we study the dynamics of highly localized excitations, or discrete breathers, which are known to be initiated by the quasistatic stretching of bonds between adjacent particles. We show via dynamical simulations that acoustic waves introduced by the harmonic term stabilize the discrete breather by suppressing the breather’s tendency to delocalize and disperse. We conclude that the harmonic term, and hence acoustic waves, are essential for the existence of localized breathers in these systems.     

From solitons to discrete breathers

The excitation of solitons and discrete breathers (pinned or otherwise, also known asintrinsic localized modes, DB/ILM) in a one-dimensional lattice, also denoted as a chain,is considered when both on-site and inter-site vibrations, coupled together, are governedby the empirical Morse interaction. We focus attention on the transformation of the formerinto the latter as the relative strength of the on-site potential to that of theinter-site potential is increased.     

Wave scattering by discrete breathers

We present a theoretical study of linear wave scattering in one-dimensional nonlinear lattices by intrinsic spatially localized dynamic excitations or discrete breathers. These states appear in various nonlinear systems and present a time-periodic localized scattering potential for plane waves. We consider the case of elastic one-channel scattering, when the frequencies of incoming and transmitted waves coincide, but the breather provides with additional spatially localized ac channels whose presence may lead to various interference patterns. The dependence of the transmission coefficient on the wave number q and the breather frequency Omega(b) is studied for different types of breathers: acoustic and optical breathers, and rotobreathers. We identify several typical scattering setups where the internal time dependence of the breather is of crucial importance for the observed transmission properties.     

Fano resonances with discrete breathers

A theoretical study of linear wave scattering by time-periodic spatially localized excitations (discrete breathers) is presented. A peculiar effect of total reflection occurs due to a Fano resonance when a localized state originating from closed channels resonates with the open channel. For the discrete nonlinear Schr?dinger chain, we give an analytical result for the frequency dependence of the transmission coefficient, including the possibility of resonant reflection. We extend the analysis to chains of weakly coupled anharmonic oscillators and discuss the relevance of the effect for electronic transport spectroscopy of mesoscopic systems.     

Energy thresholds for discrete breathers

Discrete breathers are time-periodic, spatially localized solutions of the equations of motion for a system of classical degrees of freedom interacting on a lattice. An important issue, not only from a theoretical point of view but also for their experimental detection, is their energy properties. We considerably enlarge the scenario of possible energy properties presented by Flach, Kladko, and MacKay [Phys. Rev. Lett. 78, 1207 (1997)]]. Breather energies have a positive lower bound if the lattice dimension is greater than or equal to a certain critical value dc. We show that dc can generically be greater than 2 for a large class of Hamiltonian systems. Furthermore, examples are provided for systems where discrete breathers exist but do not emerge from the bifurcation of a band edge plane wave. Some of these systems support breathers of arbitrarily low energy in any spatial dimension.     

Transverse discrete breathers in unstrained graphene

The European Physical Journal B - In this paper large resistor-capacitor (RC) networks that consist of randomly distributed conductive and capacitive elements which are much larger than those...     

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels and noise term for the anharmonic coordinates . For zero temperature, i.e. for , we prove that the support of the Fourier transform of and of the time averaged velocity-velocity correlation functions of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency we find that the energy E_T transferred to the harmonic system up to time T is proportional to T^α. If equals one of the phonon frequencies ω_ν, it is α=2. We prove that there is a zero measure set L such that for in its full measure complement R?L, it is α=0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L^′ such that for the dissipation rate is nonzero and may be subdissipative (0≤α<1) or superdissipative (1<α≤2), compared to ordinary dissipation (α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for , related to . For the small denominators do not lead to divergencies such that is a smooth and bounded function in t.     

Linear stability of breathers of the discrete NLS

We consider real breather solutions of the discrete cubic nonlinear Schrödinger equation near the limit of vanishing coupling between the lattice sites and present leading order asymptotics for the eigenvalues of the linearization around the breathers. The expansion is given in fractional powers of the intersite coupling parameter and determines the linear stability of the breathers. The method we use relies on normal form ideas and applies to one and higher-dimensional lattices. We also present some examples.     

Two-dimensional discrete gap breathers in a two-dimensional discrete diatomic Klein-Gordon lattice

We study the existence and stability of two-dimensional discrete breathers in a two-dimensionai discrete diatomic Klein-Gordon lattice consisting of alternating light and heavy atoms, with nearest-neighbor harmonic coupling. Localized solutions to the corresponding nonlinear differential equations with frequencies inside the gap of the linear wave spectrum, i.e. two-dimensional gap breathers, are investigated numerically. The numerical results of the corresponding algebraic equations demonstrate the possibility of the existence of two-dimensional gap breathers with three types of symmetries, i.e., symmetric, twin-antisymmetric and single-antisymmetric. Their stability depends on the nonlinear on-site potential （soft or hard）, the interaction potential （attractive or repulsive） and the center of the two-dimensional gap breathers （on a light or a heavy atom）.     

Moving discrete breathers in a monoatomic two-dimensional crystal

An ansatz has been proposed for setting the initial conditions in the molecular dynamics study of moving discrete breathers in monoatomic close packed crystals. The applicability of the ansatz has been demonstrated for a two-dimensional crystal with Morse interaction.     

Renormalized waves and discrete breathers in Beta-Fermi-Pasta-Ulam chains

We demonstrate via numerical simulation that in the strongly nonlinear limit the Beta-Fermi-Pasta-Ulam (Beta-FPU) system in thermal equilibrium behaves surprisingly like weakly nonlinear waves in properly renormalized normal variables. This arises because the collective effect of strongly nonlinear interactions effectively renormalizes linear dispersion frequency and leads to effectively weak interaction among these renormalized waves. Furthermore, we show that the dynamical scenario for thermalized Beta-FPU chains is spatially highly localized discrete breathers riding chaotically on spatially extended, renormalized waves.     

Different kinds of discrete breathers in a Sine-Gordon lattice

We study a one-dimensional Sine-Gordon lattice of anharmonic oscillators with cubic and quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the one-dimensional Sine-Gordon lattice no matter whether the nonlinear interaction is cubic or quartic. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.     

Classical breathers and quantum coherent states in discrete NLS systems

We present a comparison of quantum and “semiclassical” trajectories of coherent states that correspond to classical breather solutions of finite discrete nonlinear Schrödinger (DNLS) lattices. The main goal is to explain earlier numerical observations of recurrent return to the vicinity of initial coherent states corresponding to stable breathers that are also spatially localized. This effect can be considered as a quantum manifestation of classical spatial localization. We show that these phenomena are encoded in a simple expression for the distance between the quantum and semiclassical states that involves the basic frequencies of the classical and quantum systems, as well as the breather amplitude and quantum spectral decomposition of the system. A corollary is that recurrence phenomena are robust under perturbation of the initial conditions for stable breathers.     

Two-dimensional discrete breathers: construction, stability, and bifurcations

We develop a methodology for the construction of two-dimensional discrete breather excitations. Application to the discrete nonlinear Schrodinger equation on a square lattice reveals three different types of breathers. Considering an elementary plaquette, the most unstable mode is centered on the plaquette, the most stable mode is centered on its vertices, while the intermediate (but also unstable) mode is centered at the middle of one of the edges. Below the turning points of each branch in a frequency-power phase diagram, the construction methodology fails and a continuation method is used to obtain the unstable branches of the solutions until a triple point is reached. At this triple point, the branches meet and subsequently bifurcate into the final state of an extended phonon mode.     

Ab initio simulation of gap discrete breathers in strained graphene

The methods of the density functional theory were used for the first time for the simulation of discrete breathers in graphene. It is demonstrated that breathers can exist with frequencies lying in the gap of the phonon spectrum, induced by uniaxial tension of a monolayer graphene sheet in the “zigzag” direction (axis X), polarized in the “armchair” direction (axis Y). The found gap breathers are highly localized dynamic objects, the core of which is formed by two adjacent carbon atoms located on the Y axis. The atoms surrounding the core vibrate at much lower amplitudes along both the axes (X and Y). The dependence of the frequency of these breathers on amplitude is found, which shows a soft type of nonlinearity. No breathers of this type were detected in the gap induced by stretching along the Y axis. It is shown that the breather vibrations may be approximated by the Morse oscillators, the parameters of which are determined from ab initio calculations. The results are of fundamental importance, as molecular dynamics calculations based on empirical potentials cannot serve as a reliable proof of the existence of breathers in crystals.     

Modulational instability and discrete breathers in the discrete cubic-quintic nonlinear Schrödinger equation

We investigate the properties of modulational instability and discrete breathers in the cubic-quintic discrete nonlinear Schrödinger equation. We analyze the regions of modulational instabilities of nonlinear plane waves. Using the Page approach [J.B. Page, Phys. Rev. B 41 (1990) 7835], we derive the conditions for the existence and stability for bright discrete breather solutions. It is shown that the quintic nonlinearity brings qualitatively new conditions for stability of strongly localized modes. The application to the existence of localized modes in the Bose-Einstein condensate (BEC) with three-body interactions in an optical lattice is discussed. The numerical simulations agree with the analytical predictions.     

Energy exchange between discrete breathers in graphane in thermal equilibrium

Graphane is a fully hydrogenated graphene which is practically interesting for application in electronics, hydrogen storage and transportation, in nanoscale devices. As it was previously shown, the energy of a discrete breather (nonlinear localized mode) in graphane close to the value of the energy barrier at which the dehydrogenation of graphene occurs. In the present work, molecular dynamics simulation is used to investigate the possibility of energy exchange between discrete breathers in graphane in thermal equilibrium at 400 K and 600 K. In thermally equilibrated graphane, hydrogen atoms are spontaneously excited and can be considered as discrete breathers. Comparison of the kinetic energy per atom as the function of time for the selected hydrogen atoms with their displacements along the z axis showed that there is an energy exchange between the discrete breathers at evaluated temperatures. Hydrogen atom, transmitting its energy to the neighboring atom no longer exists as discrete breather. At high temperatures (600 K) the energy exchange between closely located discrete breathers also take place but strong thermo-oscillations of atoms at high temperatures (above 400 K) considerably affect the process.     

Gap discrete breathers in a two-component two-dimensional crystal in thermal equilibrium

Crystals having a gap in the phonon spectrum can maintain gap discrete breathers (DBs), i.e., nonlinear localized oscillatory modes existing in the absence of defects and having a frequency lying in the gap. The lifetime of gap DBs in a two-dimensional perfect crystal of the composition A ₃ B in thermal equilibrium has been studied by the molecular dynamics method. As was shown earlier, the existence of gap DBs in such a crystal is provided by the presence of a wide gap in the phonon spectrum if the component mass ratio m _A /m _B is sufficiently large. For comparison, a crystal with a relatively small ratio m _A /m _B is considered when the gap in the spectrum is absent and the existence of gap DBs is impossible in the case of a weak nonlinearity realized in the considered case. It has been shown that, in contrast to the opposite case, in a crystal maintaining gap DBs, long-lived localized oscillatory modes of large amplitude can emerge, whose concentration and lifetime increase with temperature.     

Intraband discrete breathers in disordered nonlinear systems. I. Delocalization

The existence of Discrete Breathers or DBs (also called Intrinsic Localized Modes or ILMs) and multibreathers, is investigated in a simple one-dimensional chain of random anharmonic oscillators with quartic potentials coupled by springs. When the breather frequency is outside and above the linearized (phonon) spectrum, the existence theorems and numerical methods previously used in periodic nonlinear models for finding time-periodic and spatially localized solutions, hold identically in random nonlinear systems. These solutions are extraband discrete breathers (EDBs). When the frequencies penetrate inside the linearized spectrum, the existence theorems do not hold. Our numerical investigations demonstrate that the strict continuation of (localized) EDBs as intraband discrete breathers (IDBs) is impossible because of cascades of bifurcations generating many discontinuities. A detailed analysis of these bifurcations for small systems with increasing sizes, shows that only a relatively small subset of the spatially extended multibreathers can be strictly continued while their frequency varies inside the phonon spectrum. We propose an ansatz for finding the coding sequences of these solutions and continuing safely these multibreathers in finite systems of any size. This continuation ends at a lower limit frequency where the solution annihilates through a bifurcation with another multibreather. A smaller subset of these multibreather solutions can be continued to amplitude zero and become linear localized modes at this limit. Conversely, any linear localized mode can be continued when increasing its frequency as an extended multibreather. Extrapolation of these results to infinite systems yields the main conclusion of this first part which is that nonlinearity in disordered systems (with localized eigenmodes only) restores their capability of energy transportation by generating infinitely many spatially extended time-periodic solutions. This approach yields mainly spatially extended solutions, except sometimes at their bifurcation points. In the second part of this work, which is presented in our next article, we develop an accurate method for calculating in situ localized IDBs.