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.     

Absence of wave packet diffusion in disordered nonlinear systems

We study the spreading of an initially localized wave packet in two nonlinear chains (discrete nonlinear Schr?dinger and quartic Klein-Gordon) with disorder. Previous studies suggest that there are many initial conditions such that the second moment of the norm and energy density distributions diverges with time. We find that the participation number of a wave packet does not diverge simultaneously. We prove this result analytically for norm-conserving models and strong enough nonlinearity. After long times we find a distribution of nondecaying yet interacting normal modes. The Fourier spectrum shows quasiperiodic dynamics. Assuming this result holds for any initially localized wave packet, we rule out the possibility of slow energy diffusion. The dynamical state could approach a quasiperiodic solution (Kolmogorov-Arnold-Moser torus) in the long time limit.     

Moving breather collisions in Klein-Gordon chains of oscillators

We investigate the collisions of moving breathers, with the same frequency, in three different Klein-Gordon chains of oscillators. The on-site potentials are: the asymmetric and soft Morse potential, the symmetric and soft sine-Gordon potential and the symmetric and hard φ⁴ potential. The simulation of a collision begins generating two identical moving breathers traveling with opposite velocities, they are obtained after perturbing two identical stationary breathers which centers are separated by a fixed number of particles. If this number is odd we obtain an on-site collision, but if this number is even we obtain an inter-site collision. Apart from this distinction, we have considered symmetric collisions, if the colliding moving breathers are vibrating in phase, and anti-symmetric collisions, if the colliding moving breathers are vibrating in anti-phase. The simulations show that the collision properties of the three chains are different. The main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities, and a stationary breather trapped at the collision region; breather generation without trapping, with the appearance of new moving breathers with opposite velocities; breather trapping at the collision region, without the appearance of new moving breathers; and breather reflection. For each Klein-Gordon chain, the collision outcomes depend on the lattice parameters, the frequency of the perturbed stationary breathers, the internal structure of the moving breathers and the number of particles that initially separates the stationary breathers when they are perturbed.     

Discrete breathers — Advances in theory and applications

Nonlinear classical Hamiltonian lattices exhibit generic solutions — discrete breathers. They are time-periodic and (typically exponentially) localized in space. The lattices have discrete translational symmetry. Discrete breathers are not confined to certain lattice dimensions. We will introduce the concept of these localized excitations and review their basic properties including dynamical and structural stability. We then focus on advances in the theory of discrete breathers in three directions — scattering of waves by these excitations, persistence of discrete breathers in long transient processes and thermal equilibrium, and their quantization. The second part of this review is devoted to a detailed discussion of recent experimental observations and studies of discrete breathers, including theoretical modelling of these experimental situations on the basis of the general theory of discrete breathers. In particular we will focus on their detection in Josephson junction networks, arrays of coupled nonlinear optical waveguides, Bose–Einstein condensates loaded on optical lattices, antiferromagnetic layered structures, PtCl based single crystals and driven micromechanical cantilever arrays.     

Energy transfer in weakly coupled nonlinear oscillator chains: Transition from a wandering breather to nonlinear self-trapping

We present analytical and numerical studies of phase-coherent dynamics of intrinsically localized excitations (breathers) in a system of two weakly coupled nonlinear oscillator chains. We show that there are two qualitatively different dynamical regimes of the coupled breathers, either immovable or slowly moving: the periodic wandering of the low-amplitude breather between the chains, and the one-chain-localization of the high-amplitude breather. These two modes of coupled breathers can be mapped exactly onto two solutions of a pendulum equation, detached by a separatrix mode. We also show that these two regimes of the coupled breathers are similar, and are described by a similar pair of equations, to the two regimes in the nonlinear tunneling dynamics of two weakly coupled Bose-Einstein condensates. On the basis of this analogy, we predict a new tunneling mode of two weakly coupled Bose-Einstein condensates in which their relative phase oscillates around π/2 modulo π.     

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）.     

Extensive Adiabatic Invariants for Nonlinear Chains

We look for extensive adiabatic invariants in nonlinear chains in the thermodynamic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to exploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrödinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invariant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates.     

Interaction of phonons with discrete breathers in one-dimensional chain with tunable type of anharmonicity

In this paper, we consider the interaction of small amplitude waves (phonons) with standing discrete breather (DB) in the one-dimensional chain of harmonically coupled particles interacting with the anharmonic one-site potential, which can be of hard-type or soft-type anharmonicity. The coefficients of phonon reflection and transmission are calculated numerically. It is found that for the case of hard-type anharmonicity (soft-type anharmonicity) DBs are more transparent for short-wavelength (long-wavelength) phonon waves, while they efficiently reflect long-wavelength (short-wavelength) phonons. In thermal equilibrium, when all phonons have equal energy density, it is found that for the same width of the transparency window, DB transmits less energy in the case of the hard-type anharmonicity. This is so because, in this case, DB reflects long-wavelength phonons, which have larger group velocity and hence greater contribution to the net energy flux through the DB. In this sense, DBs more efficiently suppress thermal conductivity in the chain with hard-type anharmonicity. Our results contribute to a better understanding of the role of discrete breathers in the heat flow in nonlinear chains.     

一维Klein-Gordon/Fermi-Pasta-Ulam混合原子链中非线性参数作用的研究

采用推广的旋转平面波近似对一维非线性Klein-Gordon/Fermi-Pasta-Ulam混合原子链的运动方程进行简化,数值求解得到该系统中存在的离散呼吸子解.研究了系统中各非线性参数对该振动模的直流、一阶简谐项和二阶简谐项三个分量的对称性的影响以及对系统中局域模的影响. 关键词： Klein-Gordon/Fermi-Pasta-Ulam混合原子链 推广的旋转平面波近似 离散呼吸子 对称性     

Standing wave instabilities, breather formation and thermalization in a Hamiltonian anharmonic lattice

Modulational instability of travelling plane waves is often considered as the first step in the formation of intrinsically localized modes (discrete breathers) in anharmonic lattices. Here, we consider an alternative mechanism for breather formation, originating in oscillatory instabilities of spatially periodic or quasiperiodic nonlinear standing waves (SWs). These SWs are constructed for Klein-Gordon or Discrete Nonlinear Schr?dinger lattices as exact time periodic and time reversible multibreather solutions from the limit of uncoupled oscillators, and merge into harmonic SWs in the small-amplitude limit. Approaching the linear limit, all SWs with nontrivial wave vectors (0 < Q < π) become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. The dynamics resulting from these instabilities is found to be qualitatively different for wave vectors smaller than or larger than π/2, respectively. In one regime persisting breathers are found, while in the other regime the system thermalizes. Received 6 October 2001 / Received in final form 1st March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: mjn@ifm.liu.se     

One-dimensional "turbulence" in a discrete lattice

We study a one-dimensional discrete analog of the von Karman flow, widely investigated in turbulence. A lattice of anharmonic oscillators is excited by both ends in order to create a large scale structure in a highly nonlinear medium, in the presence of a dissipative term proportional to the second order finite difference of the velocities, similar to the viscous term in a fluid. In a first part, the energy density is investigated in real and Fourier space in order to characterize the behavior of the system on a local scale. At low amplitude of excitation the large scale structure persists in the system but all modes are however excited and exchange energy, leading to a power law spectrum for the energy density, which is remarkably stable against changes in the model parameters, amplitude of excitation, or damping. In the spirit of shell models, this regime can be described in terms of interacting scales. At higher amplitude of excitation, the large scale structure is destroyed and the dynamics of the system can be viewed as resulting from the creation, interaction, and decay of localized excitations, the discrete breathers, the one-dimensional equivalents of vortices in a fluid. The spectrum of the energy density is well described by the spectrum of the breathers, and shows an exponential decay with the wave vector. Due to this exponential behavior, the spectrum is dominated by the most intense breathers. In this regime, the probability distribution of the increments of velocity between neighboring points is remarkably similar to the experimental results of turbulence and can be described by distributions deduced from nonextensive thermodynamics as in fluids. In a second part the power dissipated in the whole lattice is studied to characterize the global behavior of the system. Its probability distribution function shows non-Gaussian fluctuations similar to the one exhibited recently in a large class of "inertial systems," i.e., systems that cannot be divided into mesoscopic regions which are independent. The properties of the nonlinear excitations of the lattice provide a partial understanding of this behavior.     

Low-frequency breathers in polyethylene crystal

It is shown both analytically and numerically that three-dimensional low-frequency (acoustic) breathers identified as localized coupled bend and twist nonlinear waves can exist in polyethylene (PE) crystal. Their motion along the chain is accompanied by the simultaneous excitation of both (“zig” and “zag”) sub-chains forming PE macromolecule.In the region of a breather one can observe intensive out-of-plane motion of carbon atoms and relatively small displacements in the plane of the zigzag. In spite of this smallness, the “secondary” nonlinear effects turn out to be crucial for the existence of breathers.Both the existence and stability of low-frequency breathers in free motion are confirmed by computer simulation, using the Molecular Dynamics (MD) procedure. The stability of breathers with respect to thermal excitations as well as to mutual collisions and collisions with optical breathers is also discussed. We study also the breathers’ formation in different conditions. It turns out that the frequencies and extensions of the breathers can be varied in very narrow regions predicted by our analytical solution.     

Discrete breathers in classical ferromagnetic lattices with easy-plane anisotropy

Discrete breathers (nonlinear localized modes) have been shown to exist in various nonlinear Hamiltonian lattice systems. This paper is devoted to the investigation of a classical d-dimensional ferromagnetic lattice with easy plane anisotropy. Its dynamics is described via the Heisenberg model. Discrete breathers exist in such a model and represent excitations with locally tilted magnetization. They possess energy thresholds and have no analogs in the continuum limit. We are going to review the previous results on such solutions and also to report new results. Among the new results we show the existence of a big variety of these breather solutions, depending on the respective orientation of the tilted spins. Floquet stability analysis has been used to classify the stable solutions depending on their spatial structure, their frequency, and other system parameters, such as exchange interaction and local (single-ion) anisotropy.     

Invariant tori in the discrete NLS with small amplitude diffraction management

We consider the question of persistence of breather solutions of the discrete NLS equation under time-periodic perturbations corresponding to small amplitude diffraction management. The question is formulated as a problem of continuation of tori in an infinite-dimensional Hamiltonian system with symmetries and we show that one-peak breathers of the discrete NLS with zero residual diffraction can be continued to periodic or quasiperiodic solutions of the discrete NLS with small residual diffraction and small amplitude diffraction management, provided that a nonresonance condition is satisfied. We also present numerical evidence that a similar continuation should be possible for certain single-, and multi-peak breathers of the discrete NLS with small diffraction.     

Discrete nonlinear Schrödinger breathers in a phonon bath

We study the dynamics of the discrete nonlinear Schr?dinger lattice initialized such that a very long transitory period of time in which standard Boltzmann statistics is insufficient is reached. Our study of the nonlinear system locked in this non-Gibbsian state focuses on the dynamics of discrete breathers (also called intrinsic localized modes). It is found that part of the energy spontaneously condenses into several discrete breathers. Although these discrete breathers are extremely long lived, their total number is found to decrease as the evolution progresses. Even though the total number of discrete breathers decreases we report the surprising observation that the energy content in the discrete breather population increases. We interpret these observations in the perspective of discrete breather creation and annihilation and find that the death of a discrete breather cause effective energy transfer to a spatially nearby discrete breather. It is found that the concepts of a multi-frequency discrete breather and of internal modes is crucial for this process. Finally, we find that the existence of a discrete breather tends to soften the lattice in its immediate neighborhood, resulting in high amplitude thermal fluctuation close to an existing discrete breather. This in turn nucleates discrete breather creation close to a already existing discrete breather. Received 21 January 1999 and Received in final form 20 September 1999     

Continuation of normal modes in finite NLS lattices

We study the continuation of breather solutions of the discrete NLS equation as the intersite coupling parameter is varied. Considering the case of a finite one-dimensional lattice of N sites, we show the existence of N branches of breathers that persist for arbitrary coupling, thus connecting normal modes of the linear system to breathers of the uncoupled, anticontinuous limit system. The proof is based on global bifurcation theory, applied to the continuation from the weakly nonlinear regime. As the coupling parameter varies these solutions generally change their stability, and we detect parameter regions where trajectories starting near unstable breathers appear to reach equipartition of power.     

Localization of Bose-Einstein condensates in optical lattices

The dynamics of repulsive bosons condensed in an optical lattice is effectively described by the Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einstein condensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a discrete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinear Schrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled condensates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the full Bose-Hubbard model. We will show that solutions exponentially localized in space and periodic in time exist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localization phenomenon due to the interplay between bounded energy spectrum and non-linearity.     

The diffusion in the quantum Smoluchowski equation

A novel quantum Smoluchowski dynamics in an external, nonlinear potential has been derived recently. In its original form, this overdamped quantum dynamics is not compatible with the second law of thermodynamics if applied to periodic, but asymmetric ratchet potentials. An improved version of the quantum Smoluchowski equation with a modified diffusion function has been put forward in L. Machura et al. (Phys. Rev. E 70 (2004) 031107) and applied to study quantum Brownian motors in overdamped, arbitrarily shaped ratchet potentials. With this work we prove that the proposed diffusion function, which is assumed to depend (in the limit of strong friction) on the second-order derivative of the potential, is uniquely determined from the validity of the second law of thermodynamics in thermal, undriven equilibrium. Put differently, no approximation-induced quantum Maxwell demon is operating in thermal equilibrium. Furthermore, the leading quantum corrections correctly render the dissipative quantum equilibrium state, which distinctly differs from the corresponding Gibbs state that characterizes the weak (vanishing) coupling limit.     

Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators

We study the existence of travelling breathers in Klein-Gordon chains, which consist of one-dimensional networks of nonlinear oscillators in an anharmonic on-site potential, linearly coupled to their nearest neighbors. Travelling breathers are spatially localized solutions which appear time periodic in a referential in translation at constant velocity. Approximate solutions of this type have been constructed in the form of modulated plane waves, whose envelopes satisfy the nonlinear Schrödinger equation (M. Remoissenet, Phys. Rev. B 33, n.4, 2386 (1986), J. Giannoulis and A. Mielke, Nonlinearity 17, p. 551–565 (2004)). In the case of travelling waves (where the phase velocity of the plane wave equals the group velocity of the wave packet), the existence of nearby exact solutions has been proved by Iooss and Kirchgässner, who have obtained exact solitary wave solutions superposed on an exponentially small oscillatory tail (G. Iooss, K. Kirchgässner, Commun. Math. Phys. 211, 439–464 (2000)). However, a rigorous existence result has been lacking in the more general case when phase and group velocities are different. This situation is examined in the present paper, in a case when the breather period and the inverse of its velocity are commensurate. We show that the center manifold reduction method introduced by Iooss and Kirchgässner is still applicable when the problem is formulated in an appropriate way. This allows us to reduce the problem locally to a finite dimensional reversible system of ordinary differential equations, whose principal part admits homoclinic solutions to quasi-periodic orbits under general conditions on the potential. For an even potential, using the additional symmetry of the system, we obtain homoclinic orbits to small periodic ones for the full reduced system. For the oscillator chain, these orbits correspond to exact small amplitude travelling breather solutions superposed on an exponentially small oscillatory tail. Their principal part (excluding the tail) coincides at leading order with the nonlinear Schrödinger approximation.     

Long-range effects on superdiffusive algebraic solitons in anharmonic chains

Studies on thermal diffusion of lattice solitons in Fermi-Pasta-Ulam (FPU)-like lattices were recently generalized to the case of dispersive long-range interactions (LRI) of the Kac-Baker form. The variance of the soliton position shows a stronger than linear time-dependence (superdiffusion) as found earlier for lattice solitons on FPU chains with nearest-neighbour interactions (NNI). Since the superdiffusion seems to be generic for nontopological solitons, we want to illuminate the role of the soliton shape on the superdiffusive mechanism. Therefore, we concentrate on an FPU-like lattice with a certain class of power-law long-range interactions where the solitons have algebraic tails instead of the exponential tails in the case of FPU-type interactions (with or without Kac-Baker LRI).Despite of structurally similar Langevin equations which hold for the soliton position and width of the two soliton types, the algebraic solitons reach the superdiffusive long-time limit with a characteristic t^3/2 time-dependence much faster than exponential solitons. The soliton shape determines the diffusion constant in the long-time limit that is approximately a factor of π smaller for algebraic solitons. Our results appear to be generic for nonlinear excitaitons in FPU-chains, because the same superdiffusive time-dependence was also observed in simulations with discrete breathers.