Localized Breathers Research Articles

Мобильные диссипативные бризеры в цепочке нелинейных осцилляторов

In a numerical experiment, it was found that in a chain of Raleigh oscillators with a nonlinear connection between them, at least three localized stationary breathers modes may exist: non-mobile, "slow" and "fast" dissipative breathers. The dynamics of the chain elements depends on the ratio of characteristic time scales of the system which determined by the frequency of oscillators and the rigidity of the connection between them. Considered system is multistable.

Attraction centers and parity-time-symmetric delta-functional dipoles in critical and supercritical self-focusing media.

We introduce a model based on the one-dimensional nonlinear Schrödinger equation with critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against the critical or supercritical collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in nonlinear optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact analytical form. In the absence of the gain-loss term, the solitons' stability is investigated in an analytical form too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic (critical) medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to intricate alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. On the other hand, if the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers, while the collapse does not occur. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.

Multiple-time-scale analysis for pinned breathers in Bose-Hubbard chains.

Localized and pinned discrete breathers in Bose-Einstein condensates in optical lattices or in arrays of optical waveguides oscillate with frequencies which are much higher than those present in the spectrum of the background. Hence, the interaction between localized breathers and their surroundings is extremely weak leading to a multiple-time-scale perturbation expansion. We identify the leading order in the asymptotic expansion of the breather amplitude which does not average to zero after one full oscillation. The reduced model predicts a lower bound of the breather drift times and explains the topological differences between breathers in dimers, trimers, and in spatially extended one-dimensional lattices even in the presence of transport from boundary heat-baths. These analytical boundaries hold true for lattices of any length, due to the highly localized nature of breathers.

Fast Sausage Solitons and Super Nonlinearity in Coronal Loops

We consider fast sausage solitons and super nonlinearity in straight homogeneous magnetic tubes having coronal parameters. The solitonic behavior is described by the Nonlinear Schrodinger Equation-(NLSE) obtained from the ideal magneto hydrodynamic equations with suitable coronal conditions. For the first time we demonstrated that fast sausage waves are subjected to super nonlinearity and likewise introduced the super nonlinear function that defines this phenomenon. We have obtained classical localized sausage soliton and Peregrine sausage breather soliton solutions for coronal conditions. And finally we have carried out extensive numerical simulations of the evolution of a wave packet governed by the NLS equation with real nonlinear parameters and demonstrated the existence and domain of the above mentioned solitonic modes.

Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo–Miwa equation

Based on Hirota bilinear method, four kinds of localized waves, solitons, breathers, lumps and rogue waves of the extended (3+1)-dimensional Jimbo–Miwa equation are constructed. Breathers are obtained through choosing appropriate parameters on soliton solutions, while lumps and rogue waves are derived via the long wave limit on the soliton solutions. The energy, phase shift, shape, and propagation direction of these localized waves can be influenced and controlled by parameters. Considering mixed cases of the above four types of solutions, we also give many kinds of interaction solutions in the same plane with different parameters or different planes with the same parameters. Dynamical characteristics of these localized waves and interaction solutions are further analyzed and vividly demonstrated through figures.

Novel soliton breathers for the higher-order Ablowitz–Kaup–Newell–Segur hierarchy

In the framework of the nonautonomous exactly integrable higher-order equations belonging to the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy, the method for systematic construction of the bound soliton states (localized breathers on the zero background) is presented, and general and specific conditions for their appearance are formulated. We show that both all parameters of solitons forming the breather, and the varying dispersion and nonlinearities should be appropriately chosen in accordance with the exact integrability conditions of the models. Novel breather solutions of the higher-order nonlinear evolution equations with vanishing boundary conditions generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain or losses. We demonstrate that novel soliton breather solutions substantially extend the breather concept known so far. The revealed specific conditions for the breather formation can be generalized and applied to the evolution equations of the next, more and more higher orders of the AKNS hierarchy.

Do N-soliton breathers exist for the Hirota equation models?

We reveal the method for systematic construction of unexpected bound soliton states (localized breathers on the zero background) and formulate specific conditions for their appearance both in the framework of the canonical and the generalized nonautonomous Hirota models. These conditions show that both all parameters of solitons forming the breather and the dispersion and nonlinearity variations cannot be chosen independently, they are related by the exact integrability of the model. Novel N-soliton breathers derived for the nonautonomous Hirota equation with vanishing boundary conditions differ in that they generally move with varying amplitudes and velocities adapted to variations of the dispersion, nonlinearity, and gain, or losses. We apply the specific conditions for the breather formation to derive novel soliton breather solutions of the nonlinear Schrödinger and the complex modified Korteweg–de Vries equations. Consequently, we demonstrate that the way is opened to generalize and apply the presented method for other evolution equations of the Ablowitz–Kaup–Newell–Segur hierarchy.

Localized waves and interaction solutions to a (3+1)-dimensional generalized KP equation

Based on the Hirota bilinear method and long wave limit, four kinds of localized waves, namely, solitons, lumps, breathers, and rogue waves are constructed for the(3+1)-dimensional generalized KP equation. N-soliton solutions are obtained by employing bilinear method, then breathers, two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two-soliton solution and four-soliton solution. These breathers own different dynamic behaviors in the different planes. Taking a long wave limit on the two and four soliton solutions under special parameter constraints, one-order lumps and rogue waves, two-order lumps and rogue waves, and interaction solutions between lumps and rogue waves are derived. Applying the same method on the three soliton solution, interaction solutions between kink solitons with periodic solutions, lumps and rogue waves are constructed, respectively. The influence of parameters on the solution is analyzed. The propagation directions, phase shifts, energies and shapes for these solutions can be affected and controlled by the parameters. Moreover, graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions.

Modulational instability and localized breather in discrete Schrödinger equation with helicoidal hopping and a power-law nonlinearity

We investigate the discrete nonlinear Schrödinger model with helicoidal hopping and a power-law nonlinearity, motivated by the tunable nonlinearity in the model of DNA chain and ultra-cold atoms trapped in a helix-shaped optical trap. In the study of modulational instability, we find a successive destabilization along with increasing nonlinear-power. In particular, the critical amplitudes of second-stage instability decrease as nonlinear-power increases. Furthermore, it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. This link enable us to analytically estimate the critical parameters at which the breather solutions turn unstable, and find these parameters are dramatically influenced by the nonlinear-power. The stability properties of localized breathers perform an obvious change when the nonlinear power crosses a critical value γcr. It is reflected that at weak nonlinearity the breathers exhibit monostability, while exceeding γcr the bistability and instability will set in. The interplay between nonlinear-power and long-range hopping on the stability properties of breathers is also discussed in detail.

Spatiotemporal Airy Ince\u2013Gaussian wave packets in strongly nonlocal nonlinear media

The self-accelerating Airy Ince–Gaussian (AiIG) and Airy helical Ince–Gaussian (AihIG) wave packets in strongly nonlocal nonlinear media (SNNM) are obtained by solving the strongly nonlocal nonlinear Schrödinger equation. For the first time, the propagation properties of three dimensional localized AiIG and AihIG breathers and solitons in the SNNM are demonstrated, these spatiotemporal wave packets maintain the self-accelerating and approximately non-dispersion properties in temporal dimension, periodically oscillating (breather state) or steady (soliton state) in spatial dimension. In particular, their numerical experiments of spatial intensity distribution, numerical simulations of spatiotemporal distribution, as well as the transverse energy flow and the angular momentum in SNNM are presented. Typical examples of the obtained solutions are based on the ratio between the input power and the critical power, the ellipticity and the strong nonlocality parameter. The comparisons of analytical solutions with numerical simulations and numerical experiments of the AiIG and AihIG optical solitons show that the numerical results agree well with the analytical solutions in the case of strong nonlocality.

Discrete breathers in an electric lattice with an impurity: Birth, interaction, and death.

We have simulated aspects of intrinsic localized modes or discrete breathers in a modulated lumped transmission line with nonlinear varactors and a defect unit cell. As the inductance or capacitance of this cell is increased, a transition from instability to stability takes place. Namely, there exist threshold values of the inductance or capacitance of a lattice impurity for a breather to be able to attach to. A resistive defect can also anchor a breather. Moreover, by either gradually lowering all the source resistances, or else increasing the modulation frequency, multiple secondary ILMs can be spontaneously generated at host sites (with only a single inductive or capacitive defect). Further, if two impurities are subcritically spaced (the separation increasing with the amplitude of the modulation voltage), a breather can pop up midway, with no breathers at the impurity sites themselves. Finally, an ILM can pull closer its neighbors on both sides, only to perish once these ILMs have gotten sufficiently close. To our knowledge, these effects have not been reported for any discrete nonlinear system.

Discrete breathers in graphane in thermal equilibrium

Nonlinear dynamics of graphane (hydrogenated graphene) as well as some other properties of this new promising material are of high interest nowadays. One of the main challenges is the explanation of hydrogenation/dehydrogenation process of graphane at finite temperatures and the understanding of the underlying mechanisms. In present work, the hypothesis of discrete breathers working as the activators of the dehydrogenation is presented. Molecular dynamics simulation is conducted to study the discrete breathers in graphane in thermal equilibrium for temperature range 50–600 K. With the temperature increase the possibility of atom separation decreases because of thermal oscillations while the critical amplitude of the atom separation increase. It is shown that nonlinear localized modes or discrete breathers can be found in graphane at thermal equilibrium at temperature range of 400–600 K. The lifetime of discrete breather increases with the increase of its initial amplitude while temperature decrease leads to the increase of lifetime. It is concluded, that discrete breathers can facilitate the process of graphene dehydrogenation, because of their high energy.

Self-trapping of excitations: Two-dimensional quasiparticle solitons in an extended Bose-Hubbard dimer array

Considering a two-dimensional Bose-Hubbard spinor lattice with weak nearest neighbour interactions and no particle transfer between sites, we theoretically study the transport of energy from one initially excited dimer, to the rest of the lattice. Beyond a critical interaction strength, low energy on-site excitations are quickly dispersed throughout the array, while stronger excitations are self trapped, resulting in localized energy breathers and solitons. These structures are quasiparticle analogues to the discrete 2D solitons in photonic lattices. Full many-body simulations additionally demonstrate the localization of one-particle entropy.

Localized discrete breather modes in neuronal microtubules

We made an attempt to provide a realistic picture of the localization of energy in microtubules (MTs), and we intend to model the nonlinear dynamics of MTs using the “double-well” $$\phi ^4$$ form of the potential describing the dipole–dipole interactions. We investigate the modulational instability (MI) of the nonlinear plane wave solutions by considering both the wave vector (q) of the basic states and the wave vector (Q) of the perturbations as free parameters. A set of explicit criteria of MI is derived, and under the plane-wave perturbation, the constant amplitude solution becomes unstable and localized discrete breathers (DBs) solutions appear. We show numerically that MI is also an indicator of the presence of discrete breathers. We suggest that an electric field favourably leads the DB excitations towards the properly aligned end triggering a dissembly of the protofilament due to the energy release. These DBs could catalyse MT-associated proteins attachment/detachment and promote or inhibit the kinesin walk. We establish that the electromechanical vibrations in MTs can generate an electromagnetic field in the form of an electric pulse (breathers) which propagates along MT serving as signalling pathway in neuronal cells. The DBs in MT can be viewed as a bit of information whose propagation can be controlled by an electric filed. They might perform the role of elementary logic gates, thus implementing a subneuronal mode of computation. The generated DBs present us with novel possibilities for the direct interaction between the local electromagnetic field and the cytoskeletal structures in neurons. Thus, we emphasize that the effect of discreteness and electric field plays a significant role in MTs.

Two-dimensional localized Peregrine solution and breather excited in a variable-coefficient nonlinear Schrödinger equation with partial nonlocality

Hierarchies of Peregrine solution and breather solution are derived in a (2+1)-dimensional variable-coefficient nonlinear Schrodinger equation with partial nonlocality. Based on these solutions, we study the control of the excitation of Peregrine solution and breather solution in different planes. In particular, the localized Peregrine solution and breather solution are firstly reported in two-dimensional space. It is expected that our analysis and results may give new insight into higher-dimensional localized rogue waves in nonlocal media.

Stability Analysis of the Jacobian Elliptic Solutions for the Twisted Peyrard-Bishop-Dauxois Model with Solvent Interaction

We consider a twisted Peyrard-Bishop-Dauxois (PBD) model and construct the exact analytical solutions, which can describe the propagation of solitary waves by invoking a discrete Jacobian elliptic function method. These solutions include the Jacobian periodic solution as well as bubble solitons. Through the Fourier series approach, we have found that the DNA dynamics is governed by a modified discrete nonlinear Schrodinger (MDNLS) equation. A detailed analysis of the role of the twisted angle in the process of bio energy localization is presented in the form of coherent localized breather modes in a PBD model. A linear stability analysis is performed and we obtain that the stability of the solutions also depends on the twisted angle.

Modulational instability and localized breather modes in the discrete nonlinear Schrödinger equation with helicoidal hopping

We study a one-dimensional discrete nonlinear Schrödinger model with hopping to the first and a selected Nth 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.

Long-lived discrete breathers in free-standing graphene

Intrinsic localized modes or discrete breathers (DBs) are investigated by molecular dynamics simulations in free-standing graphene. DBs are generated either through thermal quenching of the graphene lattice or by proper initialization, with frequencies and lifetimes sensitively depending on the interatomic potential describing the carbon-carbon interaction. In the most realistic scenario, for which temperature-dependent molecular dynamics simulations in three dimensions using a graphene-specific interatomic potential are performed, the DBs lifetimes increase to hundreds of picoseconds even at relatively high temperatures. These lifetimes are much higher than those anticipated from earlier calculations, and may enable direct breather observation in Raman spectroscopy experiments. Our simulations provide clear estimation for the temperatures in which DBs are expected to be thermally excited (1500–2000K) representing a step forward for understanding the nonlinear physics of graphene and designing experiments in order to detect DBs, with possible impact in graphene-based future technological applications.

Large systems of discrete breathers in graphene

Nonlinear localized vibrational modes or discrete breathers are of great interest nowadays, because their role in different physical properties is still remaining unknown. Elastic strain engineering allows making quite a wide gap in the density of the phonon states of graphene, although undeformed graphene has no such a gap. Gap discrete breathers in graphene can possibly trigger defect or even crack and fracture formation and the investigation of its properties is of great importance nowadays. Clusters of discrete breathers in graphene can contribute to the considerable energy accumulation and, moreover, the energy exchange between discrete breathers was previously found. At the present work, systems composed of dozens of gap discrete breathers in graphene are investigated by molecular dynamics simulation at zero temperature. Different variations of the initial amplitudes of discrete breathers in the cluster and the combination of two initial phases are considered. It is shown that such clusters can survive for a long time during thousands oscillation periods. “Negative” initial displacement when atoms in the breather are moving towards each other is found to be more preferable in comparison with the “positive” initial displacement when atoms in the breather are moving from each other and allows excitation of the long-living breathers.

Existence and stability of intrinsic localized modes in a finite FPU chain placed in three-dimensional space

Fermi-Pasta-Ulam (FPU) chain is well known model in which energy localized vibrations exist. Such energy localized vibrations are called intrinsic localized modes (ILMs) or discrete breathers (DBs). This paper discusses existence and stability of ILMs in a finite Fermi-Pasta-Ulam chain which is placed in three-dimensional space, namely, motion of each mass is not constraint to the axis of the chain. First we derive dispersion relations of longitudinal and transversal waves. It is shown that the dispersion relations are changed with respect to the initial stretch or compression of the chain. By using Newton-Raphson method, three kinds of ILM, namely, longitudinal, transversal, and rotating modes, are found in the chain. All masses moves along the axis of the chain in longitudinal modes. The relationship between the frequency and the amplitude distribution completely coincides with ILMs in the traditional FPU chain in which motions of masses are constraint along the axis. On the other hand, main oscillations of transversal modes are perpendicular to the axis. In rotating modes, all masses rotate around the axis. Distribution of the radius of rotations are localized. Stability analysis reveals that almost all the longitudinal and the transversal ILMs are unstable. However, we found a narrow parameter region of the initial stretch in which the transversal ILM becomes stable.