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We investigate three-wave resonant interactions through both the generalized Darboux transformation method and numerical simulations. Firstly, we derive a simple multi-dark-dark-dark-soliton formula through the generalized Darboux transformation. Secondly, we use the matrix analysis method to avoid the singularity of transformed potential functions and to find the general nonsingular breather solutions. Moreover, through a limit process, we deduce the general rogue wave solutions and give a classification by their dynamics including bright, dark, four-petals, and two-peaks rogue waves. Ever since the coexistence of dark soliton and rogue wave in non-zero background, their interactions naturally become a quite appealing topic. Based on the N -fold Darboux transformation, we can derive the explicit solutions to depict their interactions. Finally, by performing extensive numerical simulations we can predict whether these dark solitons and rogue waves are stable enough to propagate. These results can be available for several physical subjects such as fluid dynamics, nonlinear optics, solid state physics, and plasma physics.

Breathers, modulation instability and recurrence phenomena are studied for the derivative nonlinear Schrodinger equation, which incorporates second order dispersion, cubic nonlinearity and self-steepening effect. By insisting on periodic boundary conditions, a cascading process will occur where initially small higher order Fourier modes can grow alongside with lower order modes. Typically a breather is first observed when all modes attain roughly the same order of magnitude. Beyond the formation of the first breather, analytical formula of spatially periodic but temporally localized breather ceases to be a valid indicator. However, numerical simulations display Fermi–Pasta–Ulam–Tsingou type recurrence. Self-steepening effect plays a crucial role in the dynamics, as it induces motion of the breather and generates chaotic behavior of the Fourier coefficients. Theoretically, correlation between breather motion and the Lax pair formulation is made. Physically, quantitative assessments of wave profile evolution are made for different initial conditions, e.g. random noise versus modulation instability mode of maximum growth rate. Potential application to fluid mechanics is discussed.

In this paper, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions.

Two integrable U ( 1 ) -invariant peakon equations are derived from the NLS hierarchy through the tri-Hamiltonian splitting method. A Lax pair, a recursion operator, a bi-Hamiltonian formulation, and a hierarchy of symmetries and conservation laws are obtained for both peakon equations. These equations are also shown to arise as potential flows in the NLS hierarchy by applying the NLS recursion operator to flows generated by space translations and U ( 1 ) -phase rotations on a potential variable. Solutions for both equations are derived using a peakon ansatz combined with an oscillatory temporal phase. This yields the first known example of a peakon breather. Spatially periodic counterparts of these solutions are also obtained.

Under investigation in this article is a multi-component AB system which models the self-induced transparency phenomenon. By using the modified Darboux transformation, we present the breather solutions of such system. We study the subtle mechanism that converts the breathing state into the solitary and periodic ones, through which we obtain various stationary nonlinear excitations such as the multi-peak solitons, (quasi) periodic waves, (quasi) anti-dark solitons, W-shaped solitons and M-shaped solitons which exhibit stationary feature. According to the analysis of the group velocity difference, we give the corresponding conversion rule and present the explicit correspondence of phase diagram of wave numbers for various converted waves, by which we show the gradient relation among these converted waves. Further, by separating the converted waves into the solitary wave as well as the periodic wave, we classify different kinds of nonlinear waves and indicate the difference of the superposition mechanism among them. We show that the breather and various converted waves are formed by different superposition modes between the solitary wave components with different localities and periodic wave components with different frequencies. By virtue of the second-order solutions, we consider all possible superposition situations of two nonlinear waves and present the corresponding nonlinear wave complexes. In particular, for the hybrid structure made of a breather and a nonlinear wave with variable velocity, we then discover that the nonlinear wave does not change its state under the conversion condition, leading to that an additional breathing structure or a dark structure is contained in the converted waves. Finally, we unveil the underlying relationship between the conversion and modulation instability.

We present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of Fermi–Pasta–Ulam (FPU) type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations. The main feature of the FPU lattices we consider is that the number of interacting neighbors varies from site to site, and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem. This property is seen in 1-D models, and in a 3-D model with a geometry obtained from protein data. The spectral analysis of these examples suggests some non-resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic Birkhoff normal forms of the FPU systems. The invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way.

We study a one-dimensional discrete nonlinear Schr\"odinger model with hopping to the first and a selected N-th neighbor, motivated by a helicoidal arrangement of lattice sites. We provide a detailed analysis of the modulational instability properties of this equation, identifying distinctive multi-stage instability cascades due to the helicoidal hopping term. Bistability is a characteristic feature of the intrinsically localized breather modes, and it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. Based on this argument, we derive analytical estimates of the critical parameters at which the fundamental on-site breather branch of solutions turns unstable. In the limit of large N, these estimates predict the emergence of an effective threshold behavior, which can be viewed as the result of a dimensional crossover to a two-dimensional square lattice., Comment: 15 pages, 9 figures

In this paper, we develop the dressing method to study the exact solutions for the vector sine-Gordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices., Comment: in Physica D: Nonlinear Phenomena (2016)

We explore the inverse scattering transforms with matrix Riemann–Hilbert problems for both focusing and defocusing modified Korteweg–de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity systematically. Using a suitable uniformization variable, the direct and inverse scattering problems are proposed on a complex plane instead of a two-sheeted Riemann surface. For the direct scattering problem, the analyticities, symmetries and asymptotic behaviors of the Jost solutions and scattering matrix, and discrete spectra are established. The inverse problems are formulated and solved with the aid of the matrix Riemann-Hilbert problems, and the reconstruction formulas, trace formulas, and theta conditions are also posed. In particular, we present the general solutions for the focusing mKdV equation with NZBCs and both simple and double poles, and for the defocusing mKdV equation with NZBCs and simple poles. Finally, some representative reflectionless potentials are in detail studied to illustrate distinct nonlinear wave structures containing solitons and breathers for both focusing and defocusing mKdV equations with NZBCs.

Various solutions to the P T -symmetric nonlocal Davey–Stewartson (DS) I equation with nonzero boundary condition are derived by constraining different tau functions of the Kadomtsev–Petviashvili hierarchy combined with the Hirota bilinear method. From the first type of tau functions of the nonlocal DS I equation, we construct: (a) general line soliton solutions sitting on either a constant background or on a background of periodic line waves and (b) general lump-type soliton solutions. We find two generic types of line solitons that we call usual line solitons and new-found ones. The usual line solitons exhibit elastic collisions, whereas the new-found ones, in the evolution process, change their waveforms from an antidark (dark) shape to a dark (antidark) one. The general lump-type soliton solutions describe the interaction between 2 N -line solitons and 2 N -lumps, which give rise to different dynamical scenarios: (i) fusion of line solitons and lumps into line solitons, (ii) fission of line solitons into lumps and line solitons, and (iii) a combination of fusion and fission processes. By constraining another type of tau functions combined with the long wave limit method, periodic line waves, rogue waves, and semi-rational solutions to the nonlocal DS I equation are obtained in terms of determinants whose matrix elements have simple algebraic expressions. Finally, different types of general solutions of the nonlocal nonlinear Schrodinger equation, namely general higher-order breathers and mixed solutions consisting of higher-order breathers and rogue waves sitting on either a constant background or on a background of periodic line waves are obtained as reductions of the corresponding solutions of the nonlocal DS I equation.

In this paper we consider a 2D hexagonal crystal lattice model first proposed by Marin, Eilbeck and Russell in 1998. We perform a detailed numerical study of nonlinear propagating localized modes, that is, propagating discrete breathers and kinks. The original model is extended to allow for arbitrary atomic interactions, and to allow atoms to travel out of the unit cell. A new on-site potential is considered with a periodic smooth function with hexagonal symmetry. We are able to confirm the existence of long-lived propagating discrete breathers. Our simulations show that, as they evolve, breathers appear to localize in frequency space, i.e. the energy moves from sidebands to a main frequency band. Our numerical findings contribute to the open question of whether exact moving breather solutions exist in 2D hexagonal layers in physical crystal lattices., Comment: Both this paper and arXiv 1408.6853 discuss similar models with the same on-site potential. This paper has a Lennard-Jones interparticle potential, 1408.6853 has a piecewise polynomial function. The latter favours the existence of long-lived kinks, and much of 1408.6853 is given to a study of these. Both models support long-lived breathers, and the present paper concentrates on such solutions

We quantize the β -Fermi–Pasta–Ulam (FPU) model with nearest and next-nearest neighbor interactions using a number conserving approximation and a numerically exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivic and Tsironis (2006), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these states by varying the system parameters. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the β -FPU model with next-nearest-neighbor interactions. We analyze the effect of the next-nearest-neighbor interactions on the eigenvalue spectrum and the correlation functions of the system.

We report numerical detection of a new type of localized structures in the frame of Majda–McLaughlin–Tabak ( M M T ) model adjusted for description of essentially nonlinear gravity waves on the surface of ideal deep water. These structures–quasibreathers or oscillating quasisolitons–can be treated as groups of freak waves closely resembling experimentally observed “Three Sisters” wave packets on the ocean surface. The M M T model has quasisolitonic solutions. Unlike N L S E solitons, M M T quasisolitons are permanently backward radiating energy, but nevertheless do exist during thousands of carrier wave periods. Quasisolitons of small amplitude are regular and stable, but large-amplitude ones demonstrate oscillations of amplitude and spectral shape. This effect can be explained by periodic formation of weak collapses, carrying out negligibly small amount of energy. We call oscillating quasisolitons “quasibreathers”.

We study deterministic escape dynamics of the discrete Klein–Gordon model with a repulsive quartic on-site potential. Using a combination of analytical techniques, based on differential and algebraic inequalities and selected numerical illustrations, we first derive conditions for collapse of an initially excited single-site unit, for both the Hamiltonian and the linearly damped versions of the system and showcase different potential fates of the single-site excitation, such as the possibility to be “pulled back” from outside the well or to “drive over” the barrier some of its neighbors. Next, we study the evolution of a uniform (small) segment of the chain and, in turn, consider the conditions that support its escape and collapse of the chain. Finally, our path from one to the few and finally to the many excited sites is completed by a modulational stability analysis and the exploration of its connection to the escape process for plane wave initial data. This reveals the existence of three distinct regimes, namely modulational stability, modulational instability without escape and, finally, modulational instability accompanied by escape. These are corroborated by direct numerical simulations. In each of the above cases, the variations of the relevant model parameters enable a consideration of the interplay of discreteness and nonlinearity within the observed phenomenology.

Combined breather–kink modes have been observed in several systems. Here we show that in the FPU lattice, small amplitude breathing–kink modes are three-soliton solutions of an associated mKdV equation. Since this is integrable, the modes can be constructed using the Backlund transform. As well as finding explicit solutions, we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others.

We study the linear stability of stationary bumps in piecewise smooth neural fields with local negative feedback in the form of synaptic depression or spike frequency adaptation. The continuum dynamics is described in terms of a nonlocal integrodifferential equation, in which the integral kernel represents the spatial distribution of synaptic weights between populations of neurons whose mean firing rate is taken to be a Heaviside function of local activity. Discontinuities in the adaptation variable associated with a bump solution means that bump stability cannot be analyzed by constructing the Evans function for a network with a sigmoidal gain function and then taking the high-gain limit. In the case of synaptic depression, we show that linear stability can be formulated in terms of solutions to a system of pseudo-linear equations. We thus establish that sufficiently strong synaptic depression can destabilize a bump that is stable in the absence of depression. These instabilities are dominated by shift perturbations that evolve into traveling pulses. In the case of spike frequency adaptation, we show that for a wide class of perturbations the activity and adaptation variables decouple in the linear regime, thus allowing us to explicitly determine stability in terms of the spectrum of a smooth linear operator. We find that bumps are always unstable with respect to this class of perturbations, and destabilization of a bump can result in either a traveling pulse or a spatially localized breather.

We study the appearance of discrete gap solitons in a nonlinear Schrodinger model with a periodic on-site potential that possesses a gap evacuated of plane-wave solutions in the linear limit. For finite lattices supporting an anti-phase (q = pi/2) gap edge phonon as an anharmonic standing wave in the nonlinear regime, gap solitons are numerically found to emerge via pitchfork bifurcations from the gap edge. Analytically, modulational instabilities between pairs of bifurcation points on this "nonlinear gap boundary" are found in terms of critical gap widths, turning to zero in the infinite-size limit, which are associated with the birth of the localized soliton as well as discrete multisolitons in the gap. Such tunable instabilities can be of relevance in exciting soliton states in modulated arrays of nonlinear optical waveguides or Bose-Einstein condensates in periodic potentials. For lattices whose gal) edge phonon only asymptotically approaches the anti-phase solution, the nonlinear gap boundary splits in a bifurcation scenario leading to the birth of the discrete gap soliton as a continuable orbit to the gap edge in the linear limit. The instability-induced dynamics of the localized soliton in the gap regime is found to thermalize according to the Gibbsian equilibrium distribution, while the spontaneous formation of persisting intrinsically localized modes (discrete breathers) from the extended out-gap soliton reveals a phase transition of the solution.

Statistical mechanics explains many localization phenomena of lattices such as the discrete nonlinear Schrodinger equation. However, numerical simulations show that the complete thermalization is rarely achieved. Instead, one observes metastable statistical states that are robust when excited locally. This paper investigates thermodynamically metastable states where the trajectory is confined to some part of the energy shell. The partition function and the entropy are computed with a perturbation method. This method is applicable to stable and metastable states, and it allows us to give approximative analytic expressions for the entropy in the complete thermodynamic state space.

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrodinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose–Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.

We study two specific features of onsite breathers in Nonlinear Schrodinger systems on d-dimensional cubic lattices with arbitrary power nonlinearity (i.e., arbitrary nonlinear exponent, n): their wavefunctions and energies close to the anti-continuum limit–small hopping limit–and their excitation thresholds. Exact results are systematically compared to the predictions of the so-called exponential ansatz (EA) and to the solution of the single nonlinear impurity model (SNI), where all nonlinearities of the lattice but the central one, where the breather is located, have been removed. In 1D, the exponential ansatz is more accurate than the SNI solution close to the anti-continuum limit, while the opposite result holds in higher dimensions. The excitation thresholds predicted by the SNI solution are in excellent agreement with the exact results but cannot be obtained analytically except in 1D. An EA approach to the SNI problem provides an approximate analytical solution that is asymptotically exact as n tends to infinity. But the EA result degrades as the dimension, d, increases. This is in contrast to the exact SNI solution which improves as n and/or d increase. Finally, in our investigation of the SNI problem we also prove a conjecture by Bustamante and Molina [C.A. Bustamante, M.I. Molina, Phys. Rev. B 62 (23) (2000) 15287] that the limiting value of the bound state energy is universal when n tends to infinity.

We investigate the properties of modulational instability and discrete breathers in the cubic–quintic discrete nonlinear Schrodinger 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.

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 [CR Acad Sci Paris 332, 581, (2001)]. We use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear Schr{\"o}dinger 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 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.

It is shown that there is an attracting quasi-periodic breather in networks of Ginzburg–Landau oscillators.

Chaotic breathers of two types are generated in two-dimensional Morse lattices with on-site harmonic potentials. In both a triangular and a square configuration, initial highest-frequency (π-mode) disturbances evolve into chaotic breathers. Depending on the magnitude of the parameters in the Morse potential, either quasi-one-dimensional localized chaotic breathers or two-dimensional localized (bell-shaped) ones are numerically observed when the on-site harmonic potential becomes significant. Both types of chaotic breather move irregularly within the lattice, and finally decay out into a thermalization state after a fairly long time. Quasi-one-dimensional localized chaotic breathers are favored as the strength of nonlinearity increases and they decay faster than two-dimensional localized ones.

In this paper a Frenkel--Kontorova model with a nonlinear interaction potential is used to describe a vacancy defect in a crystal. According to recent numerical results [Cuevas et al, Phys. Lett. A 315, 364 (2003)] the vacancy can migrate when it interacts with a moving breather. We study more thoroughly the phenomenology caused by the interaction of moving breathers with a single vacancy and also with double vacancies. We show that vacancy mobility is strongly correlated with the existence and stability properties of stationary breathers centered at the particles adjacent to the vacancy, which we will now call vacancy breathers., Comment: 11 pages, 8 figures

We present a partial review of results concerning the theory of Discrete Breathers in nonlinear hamiltonian and discrete systems. These special time-periodic solutions gained much interest during the last decade because they may be involved in numerous and various phenomena in physics and biophysics where they could produce nontrivial effects of energy focusing and transfer. We first review the principles which govern their existence and which are used in the existence proofs available up to now. Next we discuss their linear stability and the interaction of Discrete Breathers with small amplitude waves, showing also how they could grow or decay. We also briefly discuss the existence of intraband DBs in systems with a linear discrete spectrum which are not spatially periodic (random or otherwise) and where linear waves cannot propagate. We also show that nonlinearity may restore the existence of solutions that propagate energy. We review some results concerning energy transportation by DBs and especially the main features associated with their mobility. We briefly discuss new perspectives opened up by the theory for applications that, in particular, look especially interesting for biophysics.

We propose a regularized continuum model equation for describing discrete breathers or intrinsic localized modes in one-dimensional anharmonic lattices with symmetric nearest-neighbor potentials. Exact stationary breather solutions with purely hard quartic anharmonicity, as well as approximate stationary breather solutions in the general case, are found. The application of the multiple scales analysis indicates the movability of the small-amplitude breather solutions. The results of numerical simulations for the model equation fully support the analytical solutions. As regards the breather–breather collisions, the continuum model shares many common features with its discrete counterpart, which provides an opportunity to clarify the energy exchange mechanism for collisions between discrete breathers in lattices.

A kinetic equation is established for characterizing the quantum diffusion of two-vibron bound states in a molecular lattice coupled with a stochastic bath. It is shown that two trapped vibrons behave as a single particle whose coherent motion is controlled by the bound state band. The bath is responsible for the quantum decoherence of that particle and for its diffusion along the lattice. In addition, it yields a finite lifetime due to the irreversible decay of the trapped vibrons into the continuum of two-vibron free states. Diffusion, decoherence and lifetime are studied both numerically and analytically with special emphasis on the influence of intramolecular anharmonicity. It is shown that the anharmonicity prevents the diffusion of the vibron pair as a consequence of its breather-like behavior but enhances the pair lifetime. Consequently, the pair covers a finite length during its diffusion before its disappearance into the continuum of free states. The diffusion length does not depend on the anharmonicity and it is very small for low-dimensional molecular lattices.

We investigate a two-population neuronal network model of the Wilson–Cowan type with respect to existence of localized stationary solutions (“bumps”) and focus on the situation where two separate bump solutions (one narrow pair and one broad pair) exist. The stability of the bumps is investigated by means of two different approaches: The first generalizes the Amari approach, while the second is based on a direct linearization procedure. A classification scheme for the stability problem is formulated, and it is shown that the two approaches yield the same predictions, except for one notable exception. The narrow pair is generically unstable, while the broad pair is stable for small and moderate values of the relative inhibition time. At a critical relative inhibition time the broad pair is typically converted to stable breathers through a Hopf bifurcation. In our numerical example the broad pulse pair remains stable even when the inhibition time constant is three times longer than the excitation time constant. Thus, our model results do not support the claim that slow excitation mediated by, e.g., NMDA-receptors is needed to allow stable bumps.

We show the existence of localized breather solutions in an averaged version of the discrete nonlinear Schrodinger equation (NLS) with diffraction management, a system that models coupled waveguide arrays with periodic diffraction management geometries. The breather solutions are constrained extrema of the Hamiltonian of the averaged system and their existence is shown by a discrete version of the concentration-compactness principle. The main assumptions are that the averaged diffraction is sufficiently small (compared to strength of the nonlinearity) and that the sign of the nonlinearity corresponds to the focusing case. An interesting feature of the problem is that the nonlinear interaction between neighboring lattice sites can be large and is of infinite range. On the other hand, the interaction decays rapidly at sufficiently large distances, and this plays an important role in the proof. The results also apply to higher dimensional lattices, and to the discrete NLS equation.

In {\em Physica D} {\bf 91}, 223 (1996), results were obtained regarding the tangent bifurcation of the band edge modes ($q=0,\pi$) of nonlinear Hamiltonian lattices made of $N$ coupled oscillators. Introducing the concept of {\em partial isochronism} which characterises the way the frequency of a mode, $\omega$, depends on its energy, $\epsilon$, we generalize these results and show how the bifurcation energies of these modes are intimately connected to their degree of isochronism. In particular we prove that in a lattice of coupled purely isochronous oscillators ($\omega(\epsilon)$ strictly constant), the in-phase mode ($q=0$) never undergoes a tangent bifurcation whereas the out-of-phase mode ($q=\pi$) does, provided the strength of the nonlinearity in the coupling is sufficient. We derive a discrete nonlinear Schr\"odinger equation governing the slow modulations of small-amplitude band edge modes and show that its nonlinear exponent is proportional to the degree of isochronism of the corresponding orbits. This equation may be seen as a link between the tangent bifurcation of band edge modes and the possible emergence of localized modes such as discrete breathers., Comment: 23 pages, 1 figure

The earlier theory of targeted energy transfer (TET) [S. Aubry, G. Kopidakis, A.M. Morgante, G.P. Tsironis, Physica B 296 (2001) 222] between weakly coupled nonlinear oscillators (or discrete breathers) is extended to the case where the nonlinear oscillators are resonant by harmonics (Fermi resonance). Defining new detuning and coupling functions depending on the order of the resonance, it is shown that complete energy transfer may also occur by harmonics resonance under a condition similar to those of TET. This effect is called Fermi targeted energy transfer (or FTET). This theory of FTET is applied for the weakly coupled rotor-Morse oscillator system modelling selective chemical dissociation described in [A. Memboeuf, S. Aubry, preprint submitted to Physica D]. It is shown that chemical dissociation still occur by Fermi resonance at least up to resonance of order 5. Quantum FTET is also briefly discussed and shown to be related to the existence of a path of degenerate quantum states at the uncoupled limit.

A method of proving the existence of breathers in lattices of two-dimensional weakly coupled Hamiltonian oscillators is presented. This method has the advantage that provides a first approximation of the initial conditions of the breather solution together with its linear stability. Since the proof relies on the use of action-angle variables, we also present a method to perform the calculations without knowledge of the specific transformation. We illustrate the results in a lattice of quartic oscillators and in another one that consists of coupled 2D Morse oscillators. Finally, we examine the possibility of extending these results in lattices of more than two degrees of freedom per lattice site.

We study four different aspects of the generation of discrete breathers in one-(1D) and two-dimensional (2D) lattices with soft and hard nonlinearity. Breathers can be generated in thermal equilibrium and in various transient and nonequilibrium situations. We use the numerically obtained information about the spatio-temporal evolution of nonlinear lattices and combine it with analytical and heuristic results and arguments on breather existence, stability and interaction with plane waves to provide with a coherent picture of the complex nature of breather excitation in nonlinear lattices.

We prove the existence of breathers (spatially localized and time-periodic oscillations) in diatomic Fermi‐Pasta‐Ulam (FPU) chains with arbitrary mass ratio. This completes an existence result by Livi, Spicci and MacKay valid for large mass ratio. The problem is formulated as a mapping in a loop space and analyzed via a discrete spatial centre manifold reduction. Plane wave solutions of the linearized system have frequencies in a higher “optic” band or a lower “acoustic” band. For frequencies close to band edges, all small amplitude solutions of the nonlinear system lie on a finite-dimensional centre manifold, which reduces the problem locally to the study of a finite-dimensional mapping. For good parameter values, the map admits homoclinic orbits to 0 corresponding to discrete breathers. When the FPU interaction potential satisfies a hardening condition, we find breathers with frequencies slightly above the optic band, or in the gap slightly above the acoustic band. For a potential satisfying the opposite softening condition, we obtain breathers with frequencies in the gap slightly below the optic band. © 2004 Elsevier B.V. All rights reserved.

In the present work the problem of the evolution of standing and travelling breather-type waves for the two-space-dimensional Sine–Gordon equation is studied asymptotically and numerically. This work was motivated by the work of Xin [Physica D 135 (2000) 345] on modulated travelling wave solutions and the work of Piette and Zakrzewski [Nonlinearity 11 (1998) 1103] on radially symmetric, periodic standing wave solutions of the two-dimensional Sine–Gordon equation. In the present work it is shown that the dispersive radiation shed by a pulse as it evolves ultimately stabilises it and that the internal breathing motion of the pulse is intimately involved in this process. It is further shown that the linear momentum shed in radiation by a travelling pulse ultimately stabilises it. In [Nonlinearity 11 (1998) 1103] it was shown numerically that radially symmetric, periodic solutions exist for very long times. The present results describe approximately the dynamical approach to these periodic solutions, starting from distorted initial conditions. These distorted initial conditions were not considered in [Nonlinearity 11 (1998) 1103]. Moreover in the present work the stability of periodic solutions evolving from non-radially symmetric initial conditions is studied. Solutions of the approximate equations are found to be in remarkable agreement with full numerical solutions. This agreement confirms the crucial role that radiation plays in the evolution of the breathers.

We study a model of a nonlinear lattice with an on-site potential that induces a breather gap, i.e. a frequency interval where breathers are unstable. The gap is determined numerically and after thermalization of the system we compute power spectra of local displacements. We find that the gap is clearly manifested at small couplings through a large spectral contribution in the linearized phonon region that is not shifted strongly with temperature.

The existence of moving breathers is discussed in several anharmonic, atomic lattices, including the Fermi–Pasta–Ulam and Frenkel–Kontorova models. Previous work on this topic is also reviewed. It is concluded that, apart from the integrable case, moving breathers are unlikely to arise over significant time-durations in infinite, one-dimensional crystals.

The temporal Fourier spectra of stationary and slowly moving self-localized large-amplitude modes (breathers) in translationally invariant chain of coupled classical anharmonic oscillators are studied. The breathers arise naturally in the anharmonic lattice from modulational instability of short-wavelength extended vibrational modes. The frequencies of the resonant spectral peaks of the stationary breather are measured numerically and compared with an exact analytical solution for the stationary extended nonlinear sinusoidal wave in the anharmonic lattice, which is conveniently adapted. Symmetrical satellite peaks of the fundamental frequency and its higher odd harmonics in the temporal Fourier spectrum of the stationary breather are observed. Small admixture of the slowly moving breather solution to the stationary one is discussed in connection with these peaks. It is observed that the temporal Fourier spectrum of slowly moving breather consists of two main frequencies symmetrically shifted upwards and downwards with respect to the fundamental frequency of the stationary breather with the same energy, in perfect agreement with the earlier theoretical prediction of one of the authors. The relation between the group velocity of slowly moving breather and the fundamental frequency of the stationary breather with the same energy is derived.

We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, we consider Klein-Gordon (KG) chains with either soft (e.g., Morse) or hard (e.g., quartic) on-site potentials, as well as discrete nonlinear Schroedinger (DNLS) chains approximating the small-amplitude dynamics of KG chains with weak inter-site coupling. The SWs are constructed as exact time-periodic multibreather solutions from the anticontinuous limit of uncoupled oscillators. In the validity regime of the DNLS approximation these solutions can be continued into the linear phonon band, where they merge into standard harmonic SWs. For SWs with incommensurate wave vectors, this continuation is associated with an inverse transition by breaking of analyticity. When the DNLS approximation is not valid, the continuation may be interrupted by bifurcations associated with resonances with higher harmonics of the SW. Concerning the stability, we identify one class of SWs which are always linearly stable close to the anticontinuous limit. However, approaching the linear limit all SWs with nontrivial wave vectors become unstable through oscillatory instabilities, persisting for arbitrarily small amplitudes in infinite lattices. Investigating the dynamics resulting from these instabilities, we find two qualitatively different regimes for wave vectors smaller than or larger than pi/2, respectively. In one regime persisting breathers are found, while in the other regime the system rapidly thermalizes., 57 pages, 21 figures, to be published in Physica D. Revised version: Figs. 5 and 12 (f) replaced, some new results added to Sec. 5, Sec.7 (Conclusions) extended, 3 references added

The dynamics of wave groups is studied for long waves, using the framework of the extended Korteweg–de Vries equation. It is shown that the dynamics is much richer than the corresponding results obtained just from the Korteweg–de Vries equation. First, a reduction to a nonlinear Schrodinger equation is obtained for weakly nonlinear wave packets, and it is demonstrated that either the focussing or the defocussing case can be obtained. This is in contrast to the corresponding reduction for the Korteweg–de Vries equation, where only the defocussing case is obtained. Next, the condition for modulational instability is obtained. It is shown that wave packets are unstable only for a positive sign of the coefficient of the cubic nonlinear term in the extended Korteweg–de Vries equation, and for a high carrier frequency. At the boundary of this parameter space, a modified nonlinear Schrodinger equation is derived, and its steady-state solutions, including an algebraic soliton, are found. The exact breather solution of the extended Korteweg–de Vries equation is analysed. It is shown that in the limit of weak nonlinearity it transforms to a wave group with an envelope described by soliton solutions of the nonlinear Schrodinger equation and its modification as described above. Numerical simulations demonstrate the main features of wave group evolution and show some differences in the behaviour of the solutions of the extended Korteweg–de Vries equation, compared with those of the nonlinear Schrodinger equation.

We study the formation and evolution of chaotic breathers (CBs) on the Fermi–Pasta–Ulam oscillator chain with quartic nonlinearity (FPU-β system). Starting with most of the energy in a single high-frequency mode, the mode is found to breakup on a fast time scale into a number of spatially localized structures (CBs) which, on a slower time scale, coalesce into a single CB. On a usually longer time scale, depending strongly on the energy, the CB gives up its energy to lower frequency modes, approaching energy equipartition among modes. We analyze the behavior, theoretically, using an envelope approximation to the discrete chain of oscillators. For fixed boundaries, periodic nonlinear solutions are found. The numerical structures formed after the fast breakup are found to approximate the underlying equilibrium. These structures are shown, theoretically, to undergo slow translational motions, and an estimated time for them to coalesce into a single chaotic breather are found to agree with the numerically determined scaling τB∝E−1. A previously developed theory of the decay of the CB amplitude to approach equipartition is modified to explicitly consider the interaction of the breather with background modes. The scaling to equipartition of Teq∝E−2 agrees with the numerical scaling and gives the correct order of magnitude of Teq.

We introduce a general system of two coupled cubic complex Ginzburg–Landau (GL) equations that admits exact solitary-pulse (SP) solutions with a stable zero background. Besides representing a class of systems of the GL type, it also describes a dual-core nonlinear optical fiber with gain in one core and losses in the other. By means of systematic simulations, we study generic transformations of SPs in this system, which turn out to be: cascading multiplication of pulses through a subcritical Hopf bifurcation, which eventually leads to a spatio-temporal chaos; splitting of SP into stable traveling pulses; and a symmetry-breaking bifurcation transforming a standing SP into a traveling one. In some parameter region, the Hopf bifurcation is found to be supercritical, which gives rise to stable breathers. Travelling breathers are also possible in the system considered. In a certain parameter region, stable standing SPs, moving permanent-shape ones, and traveling breathers all coexist. In that case, we study collisions between various types of the pulses, which, generally, prove to be strongly inelastic.

We investigate the nonlinear response of the continuum sine-Gordon (SG) breather to an a.c. driver. We use an ansatz by Matsuda which is an exact collective variable (CV) solution for the unperturbed SG breather and uses only a single CV, r(t), which is the separation between the center of masses of the kink and antikink that make up the breather. We show that in the presence of a driver with an amplitude below the breakup threshold of the breather into kink and antikink, the a.c.-driven SG is quite accurately described by the r(t), which is a solution of an ordinary differential equation for a one-dimensional point particle in a potential V(r) driven by an a.c. driver and with an r-dependent mass, M(r). That is below the threshold for breakup, the solution for a driven r(t) and the use of the Matsuda identity gives a solution for the a.c.-driven SG, which is very close to the exact simulation of the a.c.-driven SG. We use a wavelet transform to analyze the frequency dependence of the time-dependent nonlinear response of the SG breather to the a.c. driver. We find the wavelet transforms of the CV solution and of the simulation of the a.c.-driven SG are qualitatively very similar to each other and often agree quite well quantitatively. In cases of breakup of the breather into K and A, where there is no appreciable radiation of phonons, we find the CV solution is very close to the exact simulation result.

Ten years ago, a normal form theory for the generic secondary instabilities of one-dimensional cellular patterns has been rigorously derived close to the secondary bifurcation threshold. The aim of this paper is to show that some experimental observations of localized patterns, previously unexplained, can be understood if one slightly modifies the former theory in order to take into account a linearly damped eigenmode with particular symmetry property. The model we develop, involving a huge number of unknown parameters, turns out to be quite unsatisfactory from a computational point of view. But we get round this difficulty by developing a strong analogy with the well-known problem of the propagation of a nonlinear wave in a periodic medium. With this analogy in mind, localized solutions of our model, similar to the usual conservative breather, gap or black soliton, are numerically obtained and compared with the experimental observations with quite a good qualitative agreement.

We present a linear theory for one-dimensional phonon scattering by discrete breathers in the discrete nonlinear Schrodinger equation using the transfer matrix formulation. We focus on eigenmodes in the linearized equation, which plays an important role in the scattering problem. Considering a special class of boundary conditions for both physical and unphysical eigenmodes in the non-traveling region and their continuation into the traveling region, we obtain an intuitive picture of the relation between the occurrence of perfect transmission and the localized eigenmode threshold. The perturbation approach with a transfer matrix formulation in the weak coupling limit predicts both the existence of two localized eigenmode thresholds at finite coupling strength and the structure of perfect transmission and perfect reflection. These results are shown to be applicable to a wide class of nonlinear chains including the phonon scattering problem by the discrete breather in the Klein–Gordon chain with cubic on-site potential.

We find spatially localized, time-periodic solutions (discrete breathers or DBs) in disordered nonlinear systems with frequency inside the linear phonon spectrum under conditions that strictly prohibit their existence in their periodic counterparts. For that purpose, we develop a new in situ method for the accurate calculation of these solutions which does not make use of any continuation from an anticontinuous limit. Using this method, we demonstrate that intraband localized modes (intraband discrete breathers or IDBs) at a given site with frequencies inside the discrete linear spectrum do exist, provided these frequencies do not belong to forbidden resonance gaps. Since there is a dense set of resonant frequencies, we illustrate numerically, in agreement with a theorem by Albanese and Frohlich, that the localized DBs exist provided that their frequencies belong to fat Cantor sets (i.e., with finite measure). Such a set contains as accumulation points the linear frequency of the normal mode at the occupied site. We check that many of these solutions are linearly stable and conjecture that their frequency belongs to another smaller fat Cantor set. Our numerical methods provide a much wider set of exact solutions which are multisite breathers and suggest conjectures extending the existing theorems. The physical implications of the existence of IDBs and possible applications for glasses and the persistent spectral hole burning are discussed.

This study presents various aspects of discrete breathers in diatomic FPU lattices with masses varying according to the aperiodic Fibonacci and Thue-Morse sequences. We investigate the existence of time-periodic breathers, starting from the anti-continuous limit and consider excitations of isolated light atoms, hence obtaining the domain of unique existence for these breathers. We also perform a linear stability analysis by studying the Floquet operator. The found exact solutions are used, slightly perturbed, as initial conditions for long-time simulations of the breathers. These breathers turn out to be robust. Finally we consider how initial excitations of two consecutive light atoms evolve. Depending on the properties of the phase space for the two-atom system at the anti-continuous limit, we obtain different behaviour of these excitations. Especially we find that the aperiodic lattices can support localized excitations with a continuous frequency distribution for the timescales we consider, while a periodic lattice is unable to. These excitations are referred to as chaotic breathers. (C)2000 Elsevier Science B.V. All rights reserved.

Chaotic responses induced by an applied biharmonic driven signal on the sine-Gordon (sG) system influenced by a constant dc-driven and the damping fields are investigated using a collective coordinate approach for the motion of the breather in the system. For this biharmonic signal, one term has a large amplitude at low frequency. Thus, the classical Melnikov method does not apply to such a system; however, we use the modified version of the Melnikov method to homoclinic bifurcations of the perturbed sG system. Additionally resonant breathers are studied using the modified subharmonic Melnikov theory. This dynamic behavior is illustrated by some numerical computations.