Breather States Research Articles

Interactions and collisions of discrete breathers in two-species Bose-Einstein condensates in optical lattices.

The dynamics of static and traveling breathers in two-species Bose-Einstein condensates in a one-dimensional optical lattice is modelled within the tight-binding approximation. Two coupled discrete nonlinear Schrödinger equations describe the interaction of the condensates in two cases of relevance: a mixture of two ytterbium isotopes and a mixture of (87)Rb and (41)K. Depending on their initial separation, interaction between static breathers of different species can lead to the formation of symbiotic structures and transform one of the breathers from a static into a traveling one. Collisions between traveling and static discrete breathers composed of different species are separated into four distinct regimes ranging from totally elastic when the interspecies interaction is highly attractive to mutual destruction when the interaction is sufficiently large and repulsive. We provide an explanation of the collision features in terms of the interspecies coupling and the negative effective mass of the discrete breathers.

Instability dynamics and breather formation in a horizontally shaken pendulum chain.

Inspired by the experimental results of Cuevas etal. [Phys. Rev. Lett. 102, 224101 (2009)], we consider theoretically the behavior of a chain of planar rigid pendulums suspended in a uniform gravitational field and subjected to a horizontal periodic driving force applied to the pendulum pivots. We characterize the motion of a single pendulum, finding bistability near the fundamental resonance and near the period-3 subharmonic resonance. We examine the development of modulational instability in a driven pendulum chain and find both a critical chain length and a critical frequency for the appearance of the instability. We study the breather solutions and show their connection to the single-pendulum dynamics and extend our analysis to consider multifrequency breathers connected to the period-3 periodic solution, showing also the possibility of stability in these breather states. Finally we examine the problem of breather generation and demonstrate a robust scheme for generation of on-site and off-site breathers.

Discrete breathers in honeycomb Fermi–Pasta–Ulam lattices

We consider the two-dimensional Fermi–Pasta–Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrödinger equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximized by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices.

Quantum breathers in Heisenberg ferromagnetic chains with Dzyaloshinsky-Moriya interaction.

We present an analytical study on quantum breathers in one-dimensional ferromagnetic XXZ chains with Dzyaloshinsky-Moriya interaction by means of the time-dependent Hartree approximation and the semidiscrete multiple-scale method. The stationary localized single-boson wave functions are obtained and these analytical solutions are checked by numerical simulations. With such stationary localized single-boson wave functions, we construct quantum breather states. Furthermore, the role of the Dzyaloshinsky-Moriya interaction is discussed.

Interference nature of quantum breather oscillation

In a system of coupled nonlinear oscillators, the breather (or local mode) solution is studied fully quantum mechanically, as well as by a semiclassical initial value representation of the propagator and classical Wigner dynamics. We show that the initial breather state is a superposition of almost degenerate eigenstates. From this simple observation it follows that the breather must decay and revive (i.e., oscillate with energy localization for extended times). Numerical results are shown for a two degree of freedom system. The fact that the semiclassical real-time result reproduces the full quantum one to a large degree, whereas the classical Wigner dynamics based on a similar set of trajectories does not, indicates that the breather oscillation can be viewed as an interference phenomenon.

Breathers and surface modes in oscillator chains with Hertzian interactions

We study localized waves in chains of oscillators coupled by Hertzian interactions and trapped in local potentials. This problem is originally motivated by Newton's cradle, a mechanical system consisting of a chain of touching beads subject to gravity and attached to inelastic strings. We consider an unusual setting with local oscillations and collisions acting on similar time scales, a situation corresponding e.g. to a modified Newton's cradle with beads mounted on stiff cantilevers. Such systems support static and traveling breathers with unusual properties, including double exponential spatial decay, almost vanishing Peierls-Nabarro barrier and spontaneous direction-reversing motion. We prove analytically the existence of surface modes and static breathers for anharmonic on-site potentials and weak Hertzian interactions.

New breather solutions of the model describing few-optical-cycle solitons beyond the slowly varying envelope approximation

The modified Korteweg–de Vries-sine-Gordon equation provides a general (1 + 1)-dimensional model describing few-cycle-pulse soliton propagation in two-component nonlinear media. For this model, we give its solutions in Wronskian form, which is convenient to describe interactions between solitons, breathers and their limit forms. Among these solutions, we get a stationary breather solution. Besides, the limit breather solution can provide two identical breathers traveling along the trajectories that are similar to logarithm curves rather than straight lines. As a reduction, we can also obtain solutions to the modified Korteweg–de Vries equation and the sine-Gordon equation.

Nonlinear localized modes in two-dimensional electrical lattices

We report the observation of spontaneous localization of energy in two spatial dimensions in the context of nonlinear electrical lattices. Both stationary and moving self-localized modes were generated experimentally and theoretically in a family of two-dimensional square as well as honeycomb lattices composed of 6 × 6 elements. Specifically, we find regions in driver voltage and frequency where stationary discrete breathers, also known as intrinsic localized modes (ILMs), exist and are stable due to the interplay of damping and spatially homogeneous driving. By introducing additional capacitors into the unit cell, these lattices can controllably induce mobile discrete breathers. When more than one such ILMs are experimentally generated in the lattice, the interplay of nonlinearity, discreteness, and wave interactions generates a complex dynamics wherein the ILMs attempt to maintain a minimum distance between one another. Numerical simulations show good agreement with experimental results and confirm that these phenomena qualitatively carry over to larger lattice sizes.

Discrete breathers in the Peyrard-Bishop model of DNA

The Peyrard-Bishop model, which describes the dynamics of a DNA molecule, is considered. The solutions that represent discrete breathers are derived in the framework of the model. The dynamic stability of the stationary discrete breathers with respect to small perturbations is studied. The solutions can be interpreted as the experimentally observed opening of the base pairs in the DNA double strand at the initial stages of denaturation. It is also demonstrated that the model allows the existence of mobile breathers that move in the absence of perturbations in the environment. The interaction of the mobile breathers is numerically simulated. The Peierls-Nabarro barrier and the effective mass and velocity of the breather are estimated.

Composite vortices in nonlinear circular waveguide arrays

It is known that, in continuous media, composite solitons with hidden vorticity, which are built of two mutually symmetric vortical components whose total angular momentum is zero, may be stable while their counterparts with explicit vorticity and nonzero total angular momentum are unstable. In this work, we demonstrate that the opposite occurs in discrete media: hidden vortex states in relatively small ring chains become unstable with the increase of the total power, while explicit vortices are stable, provided that the corresponding scalar vortex state is also stable. There are also stable mixed states, in which the components are vortices with different topological charges. Additionally, degeneracies in families of composite vortex modes lead to the existence of long-lived breather states which can exhibit vortex-charge flipping in one or both components.

Numerical simulation of the generation of magnetic inhomogeneities in ferromagnets with inhomogeneous parameters

The generation and evolution of magnetic inhomogeneities of the stationary breather type, which appear in a flat layer with the magnetic anisotropy and exchange interaction parameters that are different from those in the bulk of an infinite ferromagnet after transmission through a 180° domain wall, have been investigated theoretically. The dependences of the amplitude and frequency of vibrations on the parameters of the defect have been constructed for the revealed magnetic inhomogeneities, and the ranges of the parameters determining the possibility of their existence have been found.

Interaction of short single-component vector solitons

We study interaction of different-polarization single-component vector solitons of the envelope function in anisotropic media within the framework of the system of two coupled third-order nonlinear Schrodinger equations which allow for the third-order linear dispersion, nonlinear dispersion, nonlinear cross-phase modulation, and cross-nonlinear dispersion. The regimes of mutual reflection, passage, and asymptotic approach of the solitons are obtained. It is shown that the character of interaction of such solitons is determined by the initial relationship of their amplitudes and phases. The stationary mutual locations of interacting solitons and their coupled, the so-called breather states are discussed. The roles of the cubic nonlinearity, cubic cross-nonlinearity, and cross-nonlinear dispersion during interaction of solitons are studied.

Optical solitons inPT-symmetric nonlinear couplers with gain and loss

We study spatial and temporal solitons in the $\mathcal{PT}$ symmetric coupler with gain in one waveguide and loss in the other. Stability properties of the high- and low-frequency solitons are found to be completely determined by a single combination of the soliton's amplitude and the gain/loss coefficient of the waveguides. The unstable perturbations of the high-frequency soliton break the symmetry between its active and lossy components which results in a blowup of the soliton or a formation of a long-lived breather state. The unstable perturbations of the low-frequency soliton separate its two components in space blocking the power drainage of the active component and cutting the power supply to the lossy one. Eventually this also leads to the blowup or breathing.

Role of coupling of discrete breathers in split-ring-resonator-based metamaterials

Through detailed quantum calculations, the presence of classical discrete breathers and, subsequently, a two-phonon bound state (TPBS) or quantum breather (QB) state have already been shown in nonlinear photonic materials such as ferroelectrics. The latter was done in a periodic boundary condition in terms of the variations of TPBS parameters against impurity that is related to nonlinearity. Metamaterials are also nonlinear optical materials for applications as a split-ring-resonator (SRR) in antenna arrays. By using a Klein–Gordon approach, first multi-solitons and classical breathers are shown. For QBs, by using a periodic boundary condition, the variation of the TPBS parameters with coupling within the SRR elements is observed. Finally, in a non-periodic boundary condition approach, the temporal evolution of the number of quanta is shown eventually in order to derive the critical time of redistribution of quanta that is proportional to the QB's lifetime in femtoseconds, which also shows variation with coupling in the SRR system.

Low-energy vortex dynamics in the self-dual Chern–Simons–Higgs model

The relativistic Chern–Simons–Higgs theory finds application in anyonic superconductivity and contains topological vortices whose dynamics are poorly understood. The gauge fields are defined by a set of nonlinear constraint equations that can be accurately solved with effective Green’s functions, spectral methods, and a discretization scheme using lattice gauge techniques. Simulations show that low-energy two-vortex interactions are elastic with final scattering angles sensitive to vortex velocity; furthermore, vortex pairs form rotating breather states for certain impact parameters. In this study, a function that reproduces scattering angles in the adiabatic limit for nontangential collisions is presented. Simulation results are discussed in the context of analytical methods that extract vortex dynamics from low-energy effective Lagrangians, and a numerical method to calculate the effective Lagrangian is suggested. The numerical techniques used can be applied to the study of other Chern–Simon theories.

Strongly Nonlinear Beat Phenomena and Energy Exchanges in Weakly Coupled Granular Chains on Elastic Foundations

We study the dynamics of weakly interacting, strongly nonlinear one-dimensional granular chains mounted on elastic foundations. These chains are composed of a number of identical linearly elastic beads interacting with each other through Hertzian contact. No dissipative effects, such as plasticity or dry friction effects, are taken into account in our analysis. Assuming zero precompression between beads, the dynamics of the system under consideration is strongly (essentially) nonlinear, having no linear component. The complete absence of linear structural acoustics in these chains led to their characterization as “sonic vacua.” The two sources of strong nonlinearity in the considered granular chains are (i) the nonlinearizable Hertzian law interaction between adjacent beads in compression, and (ii) the possible separations between beads leading to bead collisions in the absence of compressive forces. In the current study we demonstrate that the weakly coupled granular chains possess complex dynamics leading to strong energy exchanges between them. Three different types of nonlinear beat phenomena are analytically studied, based on spatially periodic traveling waves, stationary breathers, and propagating breathers, respectively. We employ a complexification—averaging methodology that leads to smooth slow flow reduced models of the dynamics despite the discontinuous nature of the bead interactions. Verification of the derived analytical approximations with direct numerical simulations is also performed.

Quantum breathers in Klein-Gordon lattice: Non-periodic boundary condition approach

The presence of classical breathers and two-phonon bound state (TPBS) or quantum breather (QB) state through detailed quantum calculations have already been shown in technologically important ferroelectric materials, such as lithium niobate with antisite niobium charge defects concerning pinning transition, its control, and application. The latter was done in a periodic boundary condition with Bloch function in terms of significant variations of TPBS parameters against impurity, which is related to nonlinearity. In further extension of this work, in a non-periodic boundary condition and number-conserving approach, apart from various techniques available, only the temporal evolution of the number of quanta (i.e., phonons) in more sites is detailed in this present investigation for a generalized Klein-Gordon system with applications in ferroelectrics, metamaterials, and DNA. The critical time of redistribution of quanta that is proportional to the QB’s lifetime in these materials shows different types of behavior in the femtosecond range, which gives rise to the possibilities for making various devices.

Discrete breathers in Bose–Einstein condensates

Discrete breathers, originally introduced in the context of biopolymers and coupled nonlinear oscillators, are also localized modes of excitation of Bose–Einstein condensates (BEC) in periodic potentials such as those generated by counter-propagating laser beams in an optical lattice. Static and dynamical properties of breather states are analysed in the discrete nonlinear Schrödinger equation that is derived in the limit of deep potential wells, tight-binding and the superfluid regime of the condensate. Static and mobile breathers can be formed by progressive re-shaping of initial Gaussian wave-packets or by transporting atomic density towards dissipative boundaries of the lattice. Static breathers generated via boundary dissipations are determined via a transfer-matrix approach and discussed in the two analytic limits of highly localized and very broad profiles. Mobile breathers that move across the lattice are well approximated by modified analytical expressions derived from integrable models with two independent parameters: the core-phase gradient and the peak amplitude. Finally, possible experimental realizations of discrete breathers in BEC in optical lattices are discussed in the presence of residual harmonic trapping and in interferometry configurations suitable to investigate discrete breathers' interactions.

Quantum pinning-transition due to charge defects in ferroelectrics

We investigate the pinning of domain walls in ferroelectrics on the basis of the two phonon bound state (TPBS) or discrete breather state due to impurity energy levels above the phonon continua in ferroelectrics such as LiNbO3 in order to show the pinning transition, which indicates the point of easiest switching. We predict, with the help of our quantum calculations, that every ferroelectric material has such a critical point of easy switching. Here we describe the quantum origin of pinning through the findings of analytical and numerical calculations, as interpreted by a TPBS concept by such impurity or nonlinearity induced modes, by plotting various TPBS parameters against the Landau coefficient and percent impurity content. This new approach might lead to many interesting applications for device manufacturing.

Rogue Wave Observation in a Water Wave Tank

The conventional definition of rogue waves in the ocean is that their heights, from crest to trough, are more than about twice the significant wave height, which is the average wave height of the largest one-third of nearby waves. When modeling deep water waves using the nonlinear Schrödinger equation, the most likely candidate satisfying this criterion is the so-called Peregrine solution. It is localized in both space and time, thus describing a unique wave event. Until now, experiments specifically designed for observation of breather states in the evolution of deep water waves have never been made in this double limit. In the present work, we present the first experimental results with observations of the Peregrine soliton in a water wave tank.