Generation Of Breathers Research Articles

Breathers, solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrodinger equation (QNLSE) on uniform and elliptical backgrounds. The QNLSE is the general nonlinear Schrodinger equation that includes all terms up to the fifth-order dispersion. We use Darboux transformation to construct initial conditions for the dynamical generation of localized high-intensity optical waves. The condition for the breather-to-soliton conversion is provided with the analysis of soliton intensity profiles. We discover a new class of higher-order solutions in which Jacobi elliptic functions are set as background seed solutions of the QNLSE. We also introduce a method for generating a new class of rogue waves—the periodic rogue waves—based on the matching of the periodicity of higher-order breathers with the periodicity of the background dnoidal wave.

Soliton and rogue wave solutions of two-component nonlinear Schrödinger equation coupled to the Boussinesq equation**Project supported by the National Natural Science Foundation of China (Grant Nos. 11371248, 11431008, 11271254, 11428102, and 11671255) and the Fund from the Ministry of Economy and Competitiveness of Spain (Grant Nos. MTM2012-37070 and MTM2016-80276-P (AEI/FEDER,EU)).

The nonlinear Schrödinger (NLS) equation and Boussinesq equation are two very important integrable equations. They have widely physical applications. In this paper, we investigate a nonlinear system, which is the two-component NLS equation coupled to the Boussinesq equation. We obtain the bright–bright, bright–dark, and dark–dark soliton solutions to the nonlinear system. We discuss the collision between two solitons. We observe that the collision of bright–bright soliton is inelastic and two solitons oscillating periodically can happen in the two parallel-traveling bright–bright or bright–dark soliton solution. The general breather and rogue wave solutions are also given. Our results show again that there are more abundant dynamical properties for multi-component nonlinear systems.

Breathers and rogue waves: Demonstration with coupled nonlinear Schrödinger family of equations

Different types of breathers and rogue waves (RWs) are some of the important coherent structures which have been recently realized in several physical phenomena in hydrodynamics, nonlinear optics, Bose–Einstein condensates, etc. Mathematically, they have been deduced in nonlinear Schrodinger (NLS) equations. Here we show the existence of general breathers, Akhmediev breathers, Ma soliton and rogue wave solutions in coupled Manakov-type NLS equations and coupled generalized NLS equations representing four-wave mixing. We deduce their explicit forms using Hirota bilinearization procedure and bring out their exact structures and important properties. We also show the method to deduce the various breather solutions from rogue wave solutions using factorization form and the so-called imbricate series.

Instability dynamics and breather formation in a horizontally shaken pendulum chain.

Inspired by the experimental results of Cuevas etal. [Phys. Rev. Lett. 102, 224101 (2009)], we consider theoretically the behavior of a chain of planar rigid pendulums suspended in a uniform gravitational field and subjected to a horizontal periodic driving force applied to the pendulum pivots. We characterize the motion of a single pendulum, finding bistability near the fundamental resonance and near the period-3 subharmonic resonance. We examine the development of modulational instability in a driven pendulum chain and find both a critical chain length and a critical frequency for the appearance of the instability. We study the breather solutions and show their connection to the single-pendulum dynamics and extend our analysis to consider multifrequency breathers connected to the period-3 periodic solution, showing also the possibility of stability in these breather states. Finally we examine the problem of breather generation and demonstrate a robust scheme for generation of on-site and off-site breathers.

PT-symmetric coupler with a coupling defect: soliton interaction with exceptional point.

We study the interaction of a soliton in a parity-time (PT) symmetric coupler which has local perturbation of the coupling constant. This defect does not change the PT-symmetry of the system, but locally can achieve the exceptional point. We found that the symmetric solitons after interaction with the defect either transform into breathers or blow up. The dynamics of antisymmetric solitons are more complex, showing domains of successive broadening of the beam and of the beam splitting in two outward propagating solitons, in addition to the single breather generation and blowup. All the effects are preserved when the coupling strength in the center of the defect deviates from the exceptional point. If the coupling is strong enough, the only observable outcome of the soliton-defect interaction is the generation of the breather.

Dark solitons, breathers, and rogue wave solutions of the coupled generalized nonlinear Schrödinger equations.

We construct dark-dark soliton, general breather (GB), Akhmediev breather (AB), Ma soliton (MS), and rogue wave (RW) solutions of a coupled generalized nonlinear Schrödinger (CGNLS) equation. While dark-dark solitons are captured in the defocusing regime of the CGNLS system, the other solutions, namely, GB, AB, MS, and RW, are identified in the focusing regime. We also analyze the structures of GB, AB, MS, and RW profiles with respect to the four-wave mixing parameter. We show that when we increase the value of the real part of the four-wave mixing parameter, the number of peaks in the breather profile increases and the width of each peak shrinks. Interestingly, the direction of this profile also changes due to this change. As far as the RW profile is concerned the width of the peak becomes very thin when we increase the value of this parameter. Further, we consider the RW solution as the starting point, derive AB, MS, and GB in the reverse direction, and show that the solutions obtained in both directions match each other. In the course of the reverse analysis we also demonstrate how to capture the RW solutions directly from AB and MS.

Akhmediev breathers, Ma solitons, and general breathers from rogue waves: A case study in the Manakov system

We present explicit forms of general breather (GB), Akhmediev breather (AB), Ma soliton (MS), and rogue wave (RW) solutions of the two-component nonlinear Schrödinger (NLS) equation, namely Manakov equation. We derive these solutions through two different routes. In the forward route, we first construct a suitable periodic envelope soliton solution to this model from which we derive GB, AB, MS, and RW solutions. We then consider the RW solution as the starting point and derive AB, MS, and GB in the reverse direction. The second approach has not been illustrated so far for the two component NLS equation. Our results show that the above rational solutions of the Manakov system can be derived from the standard scalar nonlinear Schrödinger equation with a modified nonlinearity parameter. Through this two-way approach we establish a broader understanding of these rational solutions, which will be of interest in a variety of situations.

Nonsymmetric moving breather collisions in the Peyrard-Bishop DNA model

We study nonsymmetric collisions of moving breathers (MBs) in the Peyrard-Bishop DNA model. In this paper we have considered the following types of nonsymmetric collisions: head-on collisions of two breathers traveling with different velocities; collisions of moving breathers with a stationary trapped breather; and collisions of moving breathers traveling with the same direction. The various main observed phenomena are: one moving breather gets trapped at the collision region, and the other one is reflected; breather fusion without trapping, with the appearance of a new moving breather; and breather generation without trapping, with the appearance of new moving breathers traveling either with the same or different directions. For comparison we have included some results of a previous paper concerning to symmetric collisions, where two identical moving breathers traveling with opposite velocities collide. For symmetric collisions, the main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities and a stationary breather trapped at the collision region; and breather generation without trapping, with the appearance of new moving breathers with opposite velocities. A common feature for all types of collisions is that the collision outcome depends on the internal structure of the moving breathers and the exact number of pair-bases that initially separates the stationary breathers when they are perturbed. As some nonsymmetric collisions result in the generation of a new stationary trapped breather of larger energy, the trapping phenomenon could play an important part of the complex mechanisms involved in the initiation of the DNA transcription processes.

Moving breather collisions in Klein-Gordon chains of oscillators

We investigate the collisions of moving breathers, with the same frequency, in three different Klein-Gordon chains of oscillators. The on-site potentials are: the asymmetric and soft Morse potential, the symmetric and soft sine-Gordon potential and the symmetric and hard φ4 potential. The simulation of a collision begins generating two identical moving breathers traveling with opposite velocities, they are obtained after perturbing two identical stationary breathers which centers are separated by a fixed number of particles. If this number is odd we obtain an on-site collision, but if this number is even we obtain an inter-site collision. Apart from this distinction, we have considered symmetric collisions, if the colliding moving breathers are vibrating in phase, and anti-symmetric collisions, if the colliding moving breathers are vibrating in anti-phase. The simulations show that the collision properties of the three chains are different. The main observed phenomena are: breather generation with trapping, with the appearance of two new moving breathers with opposite velocities, and a stationary breather trapped at the collision region; breather generation without trapping, with the appearance of new moving breathers with opposite velocities; breather trapping at the collision region, without the appearance of new moving breathers; and breather reflection. For each Klein-Gordon chain, the collision outcomes depend on the lattice parameters, the frequency of the perturbed stationary breathers, the internal structure of the moving breathers and the number of particles that initially separates the stationary breathers when they are perturbed.

Nucleation of Breathers via Stochastic Resonance in Nonlinear Lattices

By applying a staggered driving force in a prototypical discrete model with a quartic nonlinearity, we demonstrate the spontaneous formation and destruction of discrete breathers with a selected frequency due to thermal fluctuations. The phenomenon exhibits the striking features of stochastic resonance: a nonmonotonic behavior as noise is increased and breather generation under subthreshold conditions. The corresponding peak is associated with a matching between the external driving frequency and the breather frequency at the average energy given by ambient temperature.

Noise induced breather generation in a sine–Gordon chain

We consider a sine–Gordon chain driven sinusoidally at one end. In the absence ofnoise, there exists a well known critical value of the amplitude beyond whichbreather modes can be generated via the phenomenon of supratransmission. Weconsider values of the driving amplitude below the critical amplitude such thatno breather propagates in the medium. We show that noise induces breathergeneration with a given probability depending on the noise intensity. We alsopropose a bifurcation diagram which extends the supratransmission effect to a morerealistic signal, namely a noisy sinusoidal excitation. We finally discuss somepromising signal processing applications that can be developed by taking intoaccount the contribution of noise in the media sharing this supratransmissionphenomenon.

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

We study collision processes 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 the possible results are: breather generation, with the formation of three new moving breathers; breather fusion, originating a new moving breather; breather trapping with also breather reflection; generation of two new moving breathers; and two new moving breathers traveling as a ligand state. Breather annihilation has never been observed.

Cutoff solitons and bistability of the discrete inductance-capacitance electrical line: Theory and experiments

A discrete nonlinear system driven at one end by a periodic excitation of frequency above the upper band edge (the discreteness induced cutoff) is shown to be a means to (1) generate propagating breather excitations in a long chain and (2) reveal the bistable property of a short chain. After detailed numerical verifications, the bistability prediction is demonstrated experimentally on an electrical transmission line made of 18 inductance-capacitance (LC) cells. The numerical simulations of the LC -line model allow us also to verify the breather generation prediction with a striking accuracy.

Breather generation in fully nonlinear models of a stratified fluid

Nonlinear wave motion is studied in a symmetric, continuously stratified, smoothed three-layer fluid in the framework of the fully nonlinear Euler equations under the Boussinesq approximation. The weakly nonlinear limit is discussed in which the governing equations can be reduced to the fully integrable modified Korteweg-de Vries equation. For some choices of the layer thicknesses the cubic nonlinear term is positive and the modified Korteweg-de Vries equation has soliton and breather solutions. Using such a stratification, the Euler equations are solved numerically using a sign-variable, initial disturbance. Breathers were generated for several forms of the initial disturbance. The breathers have moderate amplitude and to a good approximation can be described by the modified Korteweg-de Vries equation. As far as we know this is the first presentation of a breather in numerical simulations using the full nonlinear Euler equations for a stratified fluid.

Generation of solitons by a boxlike pulse in the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

The creation of solitons of the derivative nonlinear Schrödinger equation with nonvanishing boundary conditions from a boxlike initial pulse is considered. The inverse scattering transform is used. An equation for soliton eigenvalues is obtained. It is shown that simultaneous generation of breathers (solitons with internal oscillations) and one-parametric (nonoscillating) bright or dark solitons is possible. For some sets of initial parameters only breathers or only one-parametric solitons emerge. No more than three bright solitons can emerge (possibly, simultaneously with breathers) in any case, while dark solitons or breathers can arise in arbitrary number when the corresponding thresholds are exceeded. Analytical estimates for the number of generated solitons and the corresponding thresholds are given in some particular cases.

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.

On the generation of solitons and breathers in the modified Korteweg-de Vries equation.

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.

Dissipative lattice model with exact traveling discrete kink-soliton solutions: discrete breather generation and reaction diffusion regime.

We introduce a nonlinear Klein-Gordon lattice model with specific double-well on-site potential, additional constant external force and dissipation terms, which admits exact discrete kink or traveling wave fronts solutions. In the non-dissipative or conservative regime, our numerical simulations show that narrow kinks can propagate freely, and reveal that static or moving discrete breathers, with a finite but long lifetime, can emerge from kink-antikink collisions. In the general dissipative regime, the lifetime of these breathers depends on the importance of the dissipative effects. In the overdamped or diffusive regime, the general equation of motion reduces to a discrete reaction diffusion equation; our simulations show that, for a given potential shape, discrete wave fronts can travel without experiencing any propagation failure but their collisions are inelastic.

Nonlinear Adiabatic Dynamics In Polyacetylene and Related Materials

Abstract Numerical and analytical studies of the dynamic, mean-field, adiabatic Su-Schrieffer-Heeger model of polyacetylene are shown to reveal the typical importance of strong, coherent anharmonicity, e.g. as “breathers”. Examples illustrated include (i) single kink dynamics, (ii) electron-hole (e-h) decay in trans-(CH)x; (iii) e-h decay in cis-(CH)x; (iv) e-h decay in linear polyynes; (v) decay of a photoexcited kink or polaron; (vi) photoexcitation in the presence of bond or site defects. Using a variety of examples we have shown that the adiabatic dynamics of the SSH electron-phonon model typically exhibits strongly coherent anharmonic excitations, e.g. “breathing” modes. We have emphasized situations, e.g. photoexcitation, in which the energy input is high. Breather generation with spallation neutrons should also be considered, as should thermal production in materials with narrower band gaps. Persistent, localized breathers will also be typical in the presence of “confinement” mechanisms –whether du...

Generation of breathers

The inverse scattering technique is used to show that initial zero area pulses of odd symmetry with respect to time result in generating solitons of the breather type only, when they propagate in coherent absorbing media.