Properties Of Breathers Research Articles

Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers.

Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.

Two-dimensional mobile breather scattering in a hexagonal crystal lattice.

We describe the full two-dimensional scattering of long-lived breathers in a model hexagonal lattice of atoms. The chosen system, representing an idealized model of mica, combines a Lennard-Jones interatomic potential with an "egg-box" harmonic potential well surface. We investigate the dependence of breather properties on the ratio of the well depths associated with the interaction and on-site potentials. High values of this ratio lead to large spatial displacements in adjacent chains of atoms and thus enhance the two-dimensional character of the quasi-one-dimensional breather solutions. This effect is further investigated during breather-breather collisions by following the constrained energy density function in time for a set of randomly excited mobile breather solutions. Certain collisions lead to 60^{∘} scattering, and collisions of mobile and stationary breathers can generate a rich variety of states.

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.

Solitons of electric polarization in multiferroics

Complete analysis of the properties of breathers in spiral multiferroic structures in the sine-Gordon model has been presented. The methods of the excitation and detection of the breathers in the external electric and magnetic fields have been discussed.

Anisotropy of the oscillation dynamics of a breather on a trap in the electroconvective twist structure of a nematic

The oscillation dynamics of dislocations with a dissociated nucleus in the electroconvective twist structure of a nematic liquid crystal has been studied. The initial state of the roll twist structure is unstable. One of the origins of this is the helical flows of the nematic in the neighboring rolls with the antiparallel axial velocity components. As a result, oscillating or “breathing” defects are formed, which have the properties of breathers and are described by the sine-Gordon equation. It has been demonstrated that the space-time anisotropy of the character of motion of the dislocations in opposite directions orthogonal to the roll structure takes place in such a nonstationary system.

On the nonlinear stability of mKdV breathers

Breather modes of the mKdV equation on the real line are known to be elastic under collisions with other breathers and solitons. This fact indicates very strong stability properties of breathers. In this communication we describe a rigorous, mathematical proof of the stability of breathers under a class of small perturbations. Our proof involves the existence of a nonlinear equation satisfied by all breather profiles, and a new Lyapunov functional which controls the dynamics of small perturbations and instability modes. In order to construct such a functional, we work in a subspace of the energy one. However, our proof introduces new ideas in order to attack the corresponding stability problem in the energy space. Some remarks about the sine-Gordon case are also considered.

Short‐Lived Large‐Amplitude Pulses in the Nonlinear Long‐Wave Model Described by the Modified Korteweg–De Vries Equation

The appearance and disappearance of short‐lived large‐amplitude pulses in a nonlinear long wave model is studied in the framework of the modified Korteweg–de Vries equation. The major mechanism of such wave generation is modulational instability leading to the generation and interaction of the breathers. The properties of breathers are studied both within the modified Korteweg–de Vries equation, and also within the nonlinear Schrödinger equations derived by an asymptotic reduction from the modified Korteweg–de Vries for weakly nonlinear wave packets. The associated spectral problems (AKNS or Zakharov‐Shabat) of the inverse‐scattering transform technique also are utilized. Wave formation due to this modulational instability is investigated for localized and for periodic disturbances. Nonlinear‐dispersive focusing is identified as a possible mechanism for the formation of anomalously large pulses.

If "discrete breathers" is the answer, what is the question?

Intense work on discrete breathers or intrinsic localized modes in recent years has revealed a wealth of new properties of classical energy localization. Relaxation and mobility in particular may be two of the critical links with biomolecular processes. We review some of the basic discrete breather properties that we think are pertinent to biomolecules and make conjectures as to their possible biological utility.

Dynamical properties of discrete breathers in curved chains with first and second neighbor interactions.

We present the study of discrete breather dynamics in curved polymerlike chains consisting of masses connected via nonlinear springs. The polymer chains are one dimensional but not rectilinear and their motion takes place on a plane. After constructing breathers following numerically accurate procedures, we launch them in the chains and investigate properties of their propagation dynamics. We find that breather motion is strongly affected by the presence of curved regions of polymers, while the breathers themselves show a very strong resilience and remarkable stability in the presence of geometrical changes. For chains with strong angular rigidity we find that breathers either pass through bent regions or get reflected while retaining their frequency. Their motion is practically lossless and seems to be determined through local energy conservation. For less rigid chains modeled via second neighbor interactions, we find similarly that chain geometry typically does not destroy the localized breather states but, contrary to the angularly rigid chains, it induces some small but constant energy loss. Furthermore, we find that a curved segment acts as an active gate reflecting or refracting the incident breather and transforming its velocity to a value that depends on the discrete breathers frequency. We analyze the physical reasoning behind these seemingly general breather properties.

An algebraic approach to discrete breather construction

We obtain useful analytic discrete breather solutions in the form of elliptic functions for a large class of physically relevant nonlinear lattices through the application of an algebraic approach. The method we introduce is quite accurate and applies equally well to optic and acoustic chains. We present explicit results for ϕ4 and Fermi–Pasta–Ulam lattices and discuss their implications to breather properties such as generation and mobility. The method is useful also in cases where no explicit analytic solutions can be obtained. In the context of the present approach, discrete breathers are shown to be localized cnoidal modes of nonlinear lattices.

Discrete Breathers: Exact Solutions in Piecewise Linear Models

Exact breather solutions are constructed in piecewise linear (PWL) versions of the discrete nonlinear Schrodinger and Klein-Gordon equations. These solutions correspond to intersections of stable and unstable manifolds of relevant fixed points in associated 2D mappings, an exact construction of which is possible due to the PWL nature of the models. Such exact solutions give us insight into several aspects of breather properties. The problem of dynamical stability of the breathers is mentioned as an instance, detailed results on which will be presented in a future paper.

Soliton-breather reaction pathways.

We use a collective coordinate approach to investigate corpuscular properties of breathers in nonlinear lattice systems. We calculate the breather internal energy and inertial mass and use them to analyze the reaction pathways of breathers with kinks that are preformed in the lattice. We find that there is an effective kink-breather intraction potential that, under some circumstances, is attractive and has a double well shape. Furthermore, we find that in some cases the internal energy of a moving breather can be released during the reaction with the kink and subsequently transformed to kink translational energy. These breather properties seem to be model independent.

Breathers on lattices with long range interaction

We analyze the properties of breathers (time periodic spatially localized solutions) on chains in the presence of algebraically decaying interactions ${1/r}^{s}.$ We find that the spatial decay of a breather shows a crossover from exponential (short distances) to algebraic (large distances) decay. We calculate the crossover distance as a function of s and the energy of the breather. Next we show that the results on energy thresholds obtained for short range interactions remain valid for $s>3$ and that for $s<3$ (fractal dispersion at the band edge) nonzero thresholds occur for cases where the short range interaction system would yield zero threshold values.

Some stability properties of breathers in hamiltonian networks of oscillators

A one-dimensional hamiltonian chain of coupled anharmonic oscillators is considered. The averaging method is used to prove that it has breathers which are stable in ℓ 2 norm over times exponentially long with the breather's frequency. It is also shown that in a suitable approximation the breather is asymptotically stable in the sense that initial data close enough to the breather lead to solutions in which the motion of each single particle tends (as t → ∞) to the breathing oscillation. It is conjectured that also breathers of the original model are asymptotically stable.

Energy properties of discrete breathers

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degress of freedom interacting on a lattice. They come in one-parameter families. We use recent results of Flach et al. (1997) on d-dimensional systems with local interaction and recent results of Gaididei et al. (1997) on one-dimensional systems with nonlocal interaction. We discuss energy properties of breathers in d-dimensional lattices with nonlocal interactions.