Dynamics of breathers and rogue waves in scalar and multicomponent nonlinear systems

In this paper, we propose a new method, the variable separation technique, for obtaining a breather and rogue wave solution to the nonlinear evolution equation. Integrable systems of the derivative nonlinear Schrödinger type are used as three examples to illustrate the effectiveness of the presented method. We then obtain a family of rational solutions. This family of solutions includes the Akhmediev breather, the Kuznetsov-Ma breather, versatile rogue waves, and various interactions of localized waves. Moreover, the main characteristics of these solutions are discussed and some graphics are presented. More importantly, our results show that more abundant and novel localized waves may exist in the multicomponent coupled equations than in the uncoupled ones.     

Localized waves in nonlinear oscillator chains

This paper reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Fermi-Pasta-Ulam (FPU) lattice. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For FPU lattices we analyze the case when the breather period and the inverse velocity are commensurate. We employ a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation. The principal part of the reduced system is integrable and admits solutions homoclinic to quasi-periodic orbits if a hardening condition on the interaction potential is satisfied. These orbits correspond to approximate travelling breather solutions superposed on a quasi-periodic oscillatory tail. The problem of their persistence for the full system is still open in the general case. We solve this problem for an even potential if the breather period equals twice the inverse velocity, and prove in that case the existence of exact traveling breather solutions superposed on an exponentially small periodic tail.     

Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects

In this paper, we concentrate on the Hirota and Maxwell-Bloch (HMB) system governing the propagation of the femtosecond pulse in an erbium doped fiber. We present the superregular breather solutions that are nonlinear superposition of a pair of travelling breathers propagating in opposite directions. We further study the full- and half-transition modes of the superregular breather solution, which appears as a result of the coupled and higher-order effects. These waves reduce to small localized perturbations of the background at time zero.     

Breather Interaction Properties Induced by Self-Steepening and Space-Time Correction

We study the properties of breather interactions in nonlinear Kerr media with self-steepening and space-time correction and with either self-focusing or self-defocusing nonlinearity, and present a new family of exact breather solutions via the Darboux transformation with a special-designed quadratic spectral parameter. In contrast to the previous results of the nonlinear Schr?dinger equation(NLSE) hierarchy, we show that the relative phase of colliding breathers has a significant effect on the collision manifestation. In particular, only the out-of-phase interactions can generate small amplitude waves at the collision center, which are analogous to the NLSE superregular breathers. Our results will deepen our understanding of the properties of breather interactions and they will offer the possibility of experimental observations of super-regular breather dynamics in systems with self-steepening and space-time correction.     

Wave scattering by discrete breathers

We present a theoretical study of linear wave scattering in one-dimensional nonlinear lattices by intrinsic spatially localized dynamic excitations or discrete breathers. These states appear in various nonlinear systems and present a time-periodic localized scattering potential for plane waves. We consider the case of elastic one-channel scattering, when the frequencies of incoming and transmitted waves coincide, but the breather provides with additional spatially localized ac channels whose presence may lead to various interference patterns. The dependence of the transmission coefficient on the wave number q and the breather frequency Omega(b) is studied for different types of breathers: acoustic and optical breathers, and rotobreathers. We identify several typical scattering setups where the internal time dependence of the breather is of crucial importance for the observed transmission properties.     

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.     

Interaction between Breathers and Rogue Waves in a Nonlinear Optical Fiber

We study the interaction between breather and N-order rogue waves in a nonlinear optical fiber.The impacts of the relative phase and the interaction distance between breathers and rogue waves are discussed in detail.Specifically,the breather can reduce the maximum hump value of high-order rogue waves greatly in the cases of nonzero relative phase or nonzero interaction distance.The characteristic of exclusion between breathers and rogue waves is described qualitatively in the situation of different interaction distances,which can be used to change the temporal-spatial distribution of rogue waves.Their interaction properties are characterized by the trajectory of localized waves' valleys and humps.It is shown that the interaction changes the dynamical evolution trajectory of rogue waves and breathers.These results provide some possible ways to control high-order rogue waves.     

Observation of rogue wave triplets in water waves

Doubly-localised breather solutions of the nonlinear Schrödinger equation (NLS) are considered to be appropriate models to describe rogue waves in water waves as well as in other nonlinear dispersive media such as fibre optics. Within the hierarchy of this type of formations, the Peregrine breather (PB) is the lowest-order rational solution. Higher-order solutions of this kind may be understood as a nonlinear superposition of fundamental Peregrine solutions. These superpositions are nontrivial and admit only a fixed well prescribed number of elementary breathers in each higher-order solution. Here, we report first observation of second-order solution which in reality is a triplet of rogue waves.     

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.     

The Fifth Order Peregrine Breather and Its Eight-Parameter Deformations Solutions of the NLS Equation

We construct here explicitly new deformations of the Peregrine breather of order 5 with 8 real parameters. This gives new families of quasi-rational solutions of the NLS equation and thus one can describe in a more precise way the phenomena of appearance of multi rogue waves. With this method, we construct new patterns of different types of rogue waves. We get at the same time, the triangular configurations as well as rings isolated. Moreover, one sees appearing for certain values of the parameters, new configurations of concentric rings.     

Breather interactions and higher-order nonautonomous rogue waves for the inhomogeneous nonlinear Schrödinger Maxwell–Bloch equations

Under investigation in this paper are the inhomogeneous nonlinear Schrödinger Maxwell–Bloch (INLS-MB) equations which model the propagation of optical waves in an inhomogeneous nonlinear light guide doped with two-level resonant atoms. Higher-order nonautonomous breather as well as rogue wave solutions in terms of the determinants for the INLS-MB equations are presented via the n$n$-fold variable-coefficient modified Darboux transformation. The interactions among two nonautonomous breathers are graphically discussed, including the fundamental breather, bound breather, two-breather compression and two-breather evolution, etc. Moreover, several patterns of the higher-order rogue waves are also exhibited, such as the square rogue wave, two- and three-order periodic rogue waves, periodic fission and fusion, two-order stationary rogue waves, and recurrence of the two-order rogue waves. The character of the trajectory of the two-order periodic rogue wave is analyzed. Additionally, a novel type of interaction, namely, the collision between the breather and long-lived rogue waves, is found to be elastic. Our results could be useful for controlling the nonautonomous optical breathers and rogue waves in the inhomogeneous erbium doped fiber.     

Dipolar surface spin waves in antiferromagnets

We present calculations for dipolar surface spin waves in antiferromagnets at low temperatures. We calculate bulk and surface spin wave spectra for long wavelengths in the magnetostatic mode region. The different phases in an applied field along the easy axis are discussed with special emphasis on the spin flop- and paramagnetic phase. Nonreciprocal features of the surface spin waves are discussed. Numerical applications for GdAlO₃ are given.     

Fusionable and fissionable waves of (2+1)-dimensional shallow water wave equation

We investigate a (2+1)-dimensional shallow water wave equation and describe its nonlinear dynamical behaviors in physics. Based on the N-soliton solutions, the higher-order fissionable and fusionable waves, fissionable or fusionable waves mixed with soliton molecular and breather waves can be obtained by various constraints of special parameters. At the same time, by the long wave limit method, the interaction waves between fissionable or fusionable waves with higher-order lumps are acquired. Combined with the dynamic figures of the waves, the properties of the solution are deeply studied to reveal the physical significance of the waves.     

ESR investigation on the breather mode and the spinon-breather dynamical crossover in Cu benzoate

A "breather excitation" is observed directly by electron spin resonance in the quantum spin chain Cu benzoate, in which an unexpected field-induced gap has recently been found. The nonlinear field dependence of the resonance field agrees well with the formula based on a quantum sine-Gordon model. The power-law temperature dependence of the linewidth is observed in the gapless spinon regime while the width decreases exponentially for the gapped breather regime. In the intermediate range, a distinct anomaly is found, which is the manifestation of "the spinon-breather dynamical crossover."     

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.     

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.     

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.     

Soliton Molecules,Asymmetric Solitons and Hybrid Solutions for(2+1)-Dimensional Fifth-Order Kd V Equation

Soliton molecules were first discovered in optical systems and are currently a hot topic of research. We obtain soliton molecules of the(2+1)-dimensional fifth-order Kd V system under a new resonance condition called velocity resonance in theory. On the basis of soliton molecules, asymmetric solitons can be obtained by selecting appropriate parameters. Based on the N-soliton solution, we obtain hybrid solutions consisting of soliton molecules,lump waves and breather waves by partial velocity resonance and partial long wave limits. Soliton molecules,and some types of special soliton resonance solutions, are stable under the meaning that the interactions among soliton molecules are elastic. Both soliton molecules and asymmetric solitons obtained may be observed in fluid systems because the fifth-order Kd V equation describes the ion-acoustic waves in plasmas, shallow water waves in channels and oceans.     

Coexistence of triplet superconductivity and spin density waves

We discuss the possibility of the coexistence of spin density waves (antiferromagnetism) and triplet superconductivity as a particular example of a broad class of systems where the interplay of magnetism and superconductivity is important. We focus on the case of quasi-one-dimensional metals, where it is known that antiferromagnetism is in close proximity to triplet superconductivity in the pressure versus temperature phase diagram. Over a range of pressures, we propose an intermediate nonuniform phase consisting of antiferromagnetic and triplet superconducting orders. In the coexistence region, we propose a flop transition in the spin density wave order parameter vector, which affects the nature of the superconducting state and leads to the appearance of several new phases.