Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain

We find approximations to travelling breather solutions of the one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright breather and dark breather solutions are found. We find that the existence of localised (bright) solutions depends upon the coefficients of cubic and quartic terms of the potential energy, generalising an earlier inequality derived by James [G. James, Existence of breathers on FPU lattices, C. R. Acad. Sci. Paris 332 (2001) 581-586]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schrödinger equation at leading order and hence construct an asymptotic form for the breather. We show that in the absence of a cubic potential energy term, the lattice supports combined breathing-kink waveforms. The amplitude of breathing-kinks can be arbitrarily small, as opposed to the case for traditional monotone kinks, which have a nonzero minimum amplitude in such systems. We also present numerical simulations of the lattice, verifying the shape and velocity of the travelling waveforms, and confirming the long-lived nature of all such modes.     

Asymmetry of the time dynamics of breathers in the electroconvection twist structure of a nematic

Dynamics of breather defects in the periodic structures of rolls arising at electroconvection in nematic liquid crystals twisted by π/2 has been studied experimentally and theoretically. The axial components of the velocity of a hydrodynamic flow in the twist structures of nematics are opposite in the neighboring rolls. Dynamics of the breather defect is the periodic creation and annihilation of a pair of classical dislocations with the topological indices “+1” and “-1.” The annihilation occurs faster than the creation. It has been shown that the asymmetric time dynamics of the breather defect is described well by the solution of the perturbed sine-Gordon equation in the form of an interacting soliton and antisoliton.     

The sine-gordon breather as a moving oscillator in the sense of de broglie

The breather solution of the sine-Gordon equation represents an extended oscillator moving as a whole with constant velocity. We investigate the properties of the breather solution in the context of de Broglie's basic concept of “moving oscillator”, which led to quantum mechanics. We show that the momentum of the breather is proportional to its wave vector, and its total energy is proportional to the oscillator frequency.     

Bound states of breathers in the Frenkel-Kontorova model

In the dissipative, driven standard Frenkel-Kontorova model propagating breathers exist as attractors of the dynamics. In collisions, these excitations interact through the phonons they emit. A possible result of a two-breather collision is a bound state of two breathers. After looking at phonons and breather collisions, we present phenomenological results on breather bound states obtained from lattice dynamics simulations. In particular, we find that bound states can be characterised by the distance between the two breathers they comprise and their propagation velocity. Contrary to the single breather case, several values of the propagation velocity are easily accessible to bound states at fixed model parameters. The results are interpreted on the basis of the observed phonon spectra. The latter can easily be explained as Doppler-shifted combination frequencies of breather harmonics and a discreteness-induced perturbation frequency.Received: 18 December 2003, Published online: 15 March 2004PACS: 63.20.Ry Anharmonic lattice modes - 63.20.Pw Localized modes     

Emission from a small-amplitude sine-gordon breather under the action of a conservative perturbation

The power and the spectral composition of the perturbation-induced emission from a breather are calculated. It is demonstrated that the spectrum consists of separate lines, their intensity and width being exponentially small with respect to the breather's amplitude. The centers of the lines undergo the slow systematic drift.     

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.     

The confined breather: Quantized states from classical field theory

The energy of a sine-Gordon breather moving in a square-well potential is studied. The ideally reflecting walls of the well are simulated by two trains of breathers moving with opposite velocity and opposite phase, the solution being found by use of the appropriate Bäcklund transformation. The confined breather shows discrete energy levels identical with those obtained from the Schrödinger equation for a particle confined in such a potential. The breather, however, is governed by a classical, non-linear field equation for the extended field u, which is subject to classical interpretation in contrast to the statistical interpretation of the ψ-wave of quantum mechanics.     

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     

Kink-like breathers in Bose–Einstein condensates with helicoidal spin–orbit coupling

We report a kind of kink-like breathers in one-dimensional Bose–Einstein condensates (BECs) with helicoidal spin–orbit coupling (SOC), on whose two sides the background densities manifest obvious difference (called kink amplitude). The kink amplitude and shape of breather can be adjusted by the strength and period of helicoidal SOC, and its atomic number in two components exchanges periodically with time. The SOC has similar influence on the kink amplitude and the exchanged atomic number, especially when the background wave number is fixed. It indicates that the oscillating intensity of breather can be controlled by adjusting initial kink amplitude. Our work showcases the great potential of realizing novel types of breathers through SOC, and deepens our understanding on the formation mechanisms of breathers in BECs.     

Discrete gap breathers in a diatomic K2-K3-K4 chain with cubic nonlinearity

The discrete gap breathers （DGBs） in a one-dimensional diatomic chain with K2-K3-K4 potential are analysed. Using the local anharmonicity approximation, the analytical investigation has been implemented. The dependence of the central amplitude of the discrete gap breathers on the breather frequency and the localization parameter are calculated. With increasing breather frequency, the DGB amplitudes decrease. As a function of the localization parameter, the central amplitude exhibits bistability, corresponding to the two branches of the curve ω = ω（ζ）. With a nonzero cubic term, the HS mode of DGB profiles becomes weaker. With increasing K3, the HS mode of DGB profiles becomes weaker and a bit narrower. For the LS mode, with increasing K3, the central particle amplitude becomes larger, and the DGB profile becomes much sharper. But, as k3 increases further, the central particle amplitude of the LS mode becomes smaller.     

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.     

Breather trapping and breather transmission in a DNA model with an interface

We study the dynamics of moving discrete breathers in an interfaced piecewise DNA molecule. This is a DNA chain in which all the base pairs are identical and there exists an interface such that the base pairs dipole moments at each side are oriented in opposite directions. The Hamiltonian of the Peyrard-Bishop model is augmented with a term that includes the dipole-dipole coupling between base pairs. Numerical simulations show the existence of two dynamical regimes. If the translational kinetic energy of a moving breather launched towards the interface is below a critical value, it is trapped in a region around the interface collecting vibrational energy. For an energy larger than the critical value, the breather is transmitted and continues travelling along the double strand with lower velocity. Reflection phenomena never occur. The same study has been carried out when a single dipole is oriented in opposite direction to the other ones. When moving breathers collide with the single inverted dipole, the same effects appear. These results emphasize the importance of this simple type of local inhomogeneity as it creates a mechanism for the trapping of energy. Finally, the simulations show that, under favorable conditions, several launched moving breathers can be trapped successively at the interface region producing an accumulation of vibrational energy. Moreover, an additional colliding moving breather can produce a saturation of energy and a moving breather with all the accumulated energy is transmitted to the chain.     

Breather molecules and localized interaction solutions in the (2+1)-dimensional BLMP equation

The (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation is an important integrable model. In this paper, we obtain the breather molecule, the breather-soliton molecule and some localized interaction solutions to the BLMP equation. In particular, by employing a compound method consisting of the velocity resonance, partial module resonance and degeneration of the breather techniques, we derive some interesting hybrid solutions mixed by a breather-soliton molecule/breather molecule and a lump, as well as a bell-shaped soliton and lump. Due to the lack of the long wave limit, it is the first time using the compound degeneration method to construct the hybrid solutions involving a lump. The dynamical behaviors and mathematical features of the solutions are analyzed theoretically and graphically. The method introduced can be effectively used to study the wave solutions of other nonlinear partial differential equations.     

On symmetric breather collisions in lattices with saturable nonlinearity

Symmetric collisions of two discrete breathers in the lattice with saturable nonlinearity are investigated. The strong correlation of the collision properties and the parameters of colliding breathers (power, velocity, and phase difference), lattice parameters and position of the collision point is found. This is related to the internal structure of the colliding breathers and energy exchange with the phonon background. The type of collision changes from elastic to the inelastic (the breathers merging, multi-bounce interactions, breather creation etc.) with the increasing of the colliding breather power. Collision of high power breathers always results in the breather fusion. The elastic and inelastic collisions are related to the periodic and quasi-periodic colliding breathers, respectively.     

Breather Interaction Properties Induced by Self-Steepening and Space-Time Correction

We study the properties of breather interactions in nonlinear Kerr media with self-steepening and space-time correction and with either self-focusing or self-defocusing nonlinearity, and present a new family of exact breather solutions via the Darboux transformation with a special-designed quadratic spectral parameter. In contrast to the previous results of the nonlinear Schr?dinger equation(NLSE) hierarchy, we show that the relative phase of colliding breathers has a significant effect on the collision manifestation. In particular, only the out-of-phase interactions can generate small amplitude waves at the collision center, which are analogous to the NLSE superregular breathers. Our results will deepen our understanding of the properties of breather interactions and they will offer the possibility of experimental observations of super-regular breather dynamics in systems with self-steepening and space-time correction.     

Discrete moving breather collisions in a Klein-Gordon chain of oscillators

We study the collisions of moving breathers with the same frequency, traveling with opposite directions within a Klein-Gordon chain of oscillators. Two types of collisions have been analyzed: symmetric and non-symmetric, head-on collisions. For low enough frequency the outcome is strongly dependent of the dynamical states of the two colliding breathers just before the collision. For symmetric collisions, several results can be observed: breather generation, with the formation of a trapped breather and two new moving breathers; breather reflection; generation of two new moving breathers; and breather fusion bringing about a trapped breather. For non-symmetric collisions some possible results are: breather generation, with the formation of three new moving breathers; breather fusion, originating a new moving breather; breather trapping with breather reflection; generation of two new moving breathers; and two new moving breathers traveling as a bound state. Breather annihilation has never been observed.     

Comparison between Si/SiO<Subscript>2</Subscript> and InP/Al<Subscript>2</Subscript>O<Subscript>3</Subscript> based MOSFETs

The discrete breathers in graphane in thermodynamic equilibrium in the temperature range 50–600 K are studied by molecular dynamics simulation. A discrete breather is a hydrogen atom vibrating along the normal to a sheet of graphane at a high amplitude. As was found earlier, the lifetime of a discrete breather at zero temperature corresponds to several tens of thousands of vibrations. The effect of temperature on the decay time of discrete breathers and the probability of their detachment from a sheet of graphane are studied in this work. It is shown that closely spaced breathers can exchange energy with each other at zero temperature. The data obtained suggest that thermally activated discrete breathers can be involved in the dehydrogenation of graphane, which is important for hydrogen energetics.     

Breather-induced quantised superfluid vortex filaments and their characterisation

We study and characterise the breather-induced quantised superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an otherwise perturbed helical vortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex. The loop induced by the Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by the super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather. In particular, we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers. Despite the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size. These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices.     

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.     

Breather Dynamics of the Sine-Gordon Equation

This paper studies the adiabatic dynamics of the breather soliton of the sine-Gordon equation. The integrals of motion are found and then used in soliton perturbation theory to derive the differential equation governing the soliton velocity. Time-dependent functions arise and their properties are studied. These functions are found to be bounded and periodic and affect the soliton velocity. The soliton velocity is numerically plotted against time for different combinations of initial velocities and perturbation terms.