Experiments on the interactions between impurities and solitary waves in lattice model

The interactions between solitary waves and impurities have been studied experimentally in a 1D nonlinear coupled pendulum chain under vertical excitation. The mass and the coupling are unique, except a single pendulum with length impurity in the chain. The experiment reveals: the long impurity repels breather and attracts kink while the short one attracts breather and repels kink under higher frequency driving, and the long impurity attracts breather and repels kink while the short one repels breather and attracts kink under the lower frequency driving. These results prove the current theoretical prediction based on continuum-limit approximation.     

Anomalous decay and breather formation in doped alkali halides

Anomalous decay of doped alkali halides has been ascribed to breather formation in the immediate neighborhood of the impurity. New results support this connection. We report experimental data for NaBr and RbBr crystals showing anomalies in their slow emission decay. These data complete the earlier picture, confirming that the decay anomaly becomes bigger as the host-lattice-anion/cation mass ratio increases. We show the correlation between the decay anomaly and the breather formation as a function of this ratio. By simulating finite-temperature effects (which do not just involve white noise) in the lattice dynamics, we demonstrate that with increasing temperature the breather weakens, as is experimentally observed for the decay anomaly.     

Breather‐to‐soliton transitions and nonlinear wave interactions for the nonlinear Schrödinger equation with the sextic operators in optical fibers

We find that the sextic nonlinear Schrödinger (NLS) equation admits breather‐to‐soliton transitions. With the Darboux transformation, analytic breather solutions with imaginary eigenvalues up to the second order are explicitly presented. The condition for breather‐to‐soliton transitions is explicitly presented and several examples of transitions are shown. Interestingly, we show that the sextic NLS equation admits not only the breather‐to‐bright‐soliton transitions but also the breather‐to‐dark‐soliton transitions. We also show the interactions between two solitons on the constant backgrounds, as well as between breather and soliton.     

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.     

Mutual-focusing of two co-propagating beams and formation of trapped spatial breather pair in saturable nonlinear media

In this paper, using parabolic equation approach, coupled propagation of two coaxially co-propagating and mutually incoherent bright cylindrical beams in saturable nonlinear medium has been investigated. Considering the coupling coefficient equal to unity (κ=1), a detailed account of formation of spatial soliton pair (i.e. both beams are stationary trapped) and spatial breather pair (i.e. width of each beam oscillates with the propagation distance) has been provided and existence of spatially trapped breather pair (i.e. average width of each breather of the pair does not change with the propagation distance) has been shown. Conditions of formation of trapped spatial breather pair and their existence line has also been revealed for arbitrary beam width ratio of the beams. It is revealed that spatial soliton pairs are just a special case of trapped breather pair. The regions (conditions) of mutual-focusing and mutual-defocusing of spatial soliton pair/breather pair have also been identified. Lastly, the law of trapped breather pair formation is proposed.     

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     

Evolution of a randomly modulated breather in a nonlinear medium

The evolution of a randomly modulated sine-Gordon breather in a nonlinear medium is studied theoretically. The initial wave field is affected by multiplicative noise. For breather amplitude and velocity, the probability distribution function is determined by means of the inverse scattering transform and the method of cumulants. The distributions are shown to be non-Gaussian. The mean and the most probable values of the breather amplitude and velocity are calculated.     

Modulation of the breather frequency in the ac-driven sine-Gordon system with loss

In the sine-Gordon system with loss a breather can be maintained in a steady state if an ac-force of sufficient strength is applied. A small disturbance of this steady state results in a modulation of the breather frequency. The frequency of the modulation is much smaller than the breather frequency and the modulation eventually decays. This modulation frequency is determined by a perturbation method and compared with numerical experiments. Excellent agreement is found.     

Breather and double-pole solutions of the derivative nonlinear Schrödinger equation from optical fibers

The derivative nonlinear Schrödinger (DNLS) equation, which governs the propagation of the femtosecond optical pulse in a monomodal optical fiber, is analytically studied in this Letter. Breather and double-pole solutions are derived from the two-soliton solution with the choice of parameters. It is found that the parameters in the DNLS equation cannot only control the phase and propagation direction of the breather and double pole, but also influence the interaction period of the breather. Elastic collisions between a breather and a dark/anti-dark soliton are studied by the qualitative analysis and graphical illustration. The stability of the breather and double-pole solutions is also analyzed.     

Resonant breather states in Josephson coupled systems

A review of diverse resonant effects appearing in weakly dissipative Josephson coupled systems in the presence of inhomogeneous dynamic localized state (discrete breather) is given. As particular examples I discuss the resonant interaction of breather states with linear electromagnetic excitations (EEs) in dc driven Josephson junction ladders and a single plaquette containing three Josephson junctions. Such resonant interaction manifests itself by resonant steps and various sharp switchings (voltage jumps) in the current-voltage characteristics. Moreover, the resonant interaction leads to an increase of breather dynamical complexity, e.g., enlargement of the breather core, low symmetry or quasiperiodic breather states. I show that the application of an external magnetic field allows to tune the resonant interaction, and correspondingly to increase (or decrease) the height of the resonant steps, to change the stability of the breather states.     

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.     

Formation mechanism of asymmetric breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations

Based on the developed Darboux transformation, we investigate the exact asymmetric solutions of breather and rogue waves in pair-transition-coupled nonlinear Schr?dinger equations. As an example, some types of exact breather solutions are given analytically by adjusting the parameters. Moreover, the interesting fundamental problem is to clarify the formation mechanism of asymmetry breather solutions and how the particle number and energy exchange between the background and soliton ultimately form the breather solutions. Our results also show that the formation mechanism from breather to rogue wave arises from the transformation from the periodic total exchange into the temporal local property.     

Twenty Parameters Families of Solutions to the NLS Equation and the Eleventh Peregrine Breather

The Peregrine breather of order eleven(P_(11) breather) solution to the focusing one-dimensional nonlinear Schrdinger equation(NLS) is explicitly constructed here. Deformations of the Peregrine breather of order 11 with 20 real parameters solutions to the NLS equation are also given: when all parameters are equal to 0 we recover the famous P_(11) breather. We obtain new families of quasi-rational solutions to the NLS equation in terms of explicit quotients of polynomials of degree 132 in x and t by a product of an exponential depending on t. We study these solutions by giving patterns of their modulus in the(x; t) plane, in function of the different parameters.     

Breather and double-pole solutions for the Benjamin-Ono equation in a stratified fluid

The Benjamin–Ono equation is hereby investigated, which arises in the context of long internal gravity waves in a stratified fluid. With the Hirota method and symbolic computation, breather solutions are derived. Propagation of the breather and elastic collisions between the breather and soliton are graphically analyzed. The collision period and the bunch number in a wave packet are relevant to the ratio of the real part to the imaginary of the wavenumber. Through the coalescence of wavenumbers in the two-soliton solutions, we obtain the double-pole solutions.     

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.     

强非局域非线性介质中强光导引的弱光呼吸子传输规律研究

研究了强光导引的弱光呼吸子传输问题. 尽管弱光本身的非线性效应可以忽略, 但强光通过强非局域非线性效应会对弱光形成约束并与弱光本身的衍射效应达到动态平衡, 从而形成呼吸子. 得到了弱光呼吸子的解析表达式, 在此基础上分析了其束宽的呼吸演化规律, 并对弱光呼吸子整体轨迹的形状、方位、旋向等性质进行了系统的研究.     

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     

Discrete soliton collisions in a waveguide array with saturable nonlinearity

We study the symmetric collisions of two mobile breathers/solitons in a model for coupled wave guides with a saturable nonlinearity. The saturability allows the existence of mobile breathers with high power. Three main regimes are observed: breather fusion, breather reflection and breather creation. The last regime seems to be exclusive of systems with a saturable nonlinearity, and has been previously observed in continuous models. In some cases a “symmetry breaking” can be observed, which we show to be an numerical artifact.     

Higher-order smooth positons and breather positons of Sine-Gordon equation

According to the N-soliton solution derived from Hirota’s bilinear method, higher-order smooth positons and breather positons are obtained efficiently through an ingenious limit approach. This paper takes the Sine-Gordon equation as an example to introduce how to utilize this technique to generate these higher-order smooth positons and breather positons in detail. The dynamical behaviors of smooth positons and breather positons are presented by some figures. During the procedure of deduction, the approach mentioned has the strengths of concision and celerity. In terms of feasibility and practicability, this approach can be exploited widely to study higher-order smooth positons and breather positons of other integrable systems.     

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.