Interaction of longitudinal phonons with discrete breather in strained graphene

We numerically analyze the interaction of small-amplitude phonon waves with standing gap discrete breather (DB) in strained graphene. To make the system support gap DB, strain is applied to create a gap in the phonon spectrum. We only focus on the in-plane phonons and DB, so the issue is investigated under a quasi-one-dimensional setup. It is found that, for the longitudinal sound waves having frequencies below 6 THz, DB is transparent and thus no radiation of energy from DB takes place; whereas for those sound waves with higher frequencies within the acoustic (optical) phonon band, phonon is mainly transmitted (reflected) by DB, and concomitantly, DB radiates its energy when interacting with phonons. The latter case is supported by the fact that, the sum of the transmitted and reflected phonon energy densities is noticeably higher than that of the incident wave. Our results here may provide insight into energy transport in graphene when the spatially localized nonlinear vibration modes are presented.     

New equation for the description of inelastic interaction of nonlinear localized waves in dispersive media

A wave equation for the simulation of nonlinear plane solitary perturbations of the free surface of a shallow fluid has been derived. In contrast to the modified Boussinesq equation, the new one correctly describes the interaction of counter-propagating small-amplitude waves. It has been shown analytically that collisions of solitons are inelastic even in the first-order perturbation theory and the nonlinear dynamics of such collisions is qualitatively different from that described by the modified Boussinesq equation.     

Vector breather of surface plasmon polaritons in left-handed material

A theory of an optical vector breather of self-induced transparency of surface plasmon polaritons is constructed. The nonlinear surface TM-modes propagating along the interface separating an isotropic medium and anisotropic left-handed material are investigated. A transition layer sandwiched between the connected media is described using a model of a two-dimensional gas of semiconductor quantum dots. Explicit analytical expressions for the shape and parameters of the surface vector breathers are obtained as well as simulations of the space-time dynamics of two-dimensional vector breathers presented with realistic parameters which can be reached in current experiments. It is shown that properties of a surface vector breather with phase modulation depends on the parameters of the quantum dots, the connected media and the transverse structure of the surface wave.     

Inclined periodic homoclinic breather and rogue waves for the (1+1)-dimensional Boussinesq equation

A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (1+1)-dimensional Boussinesq equation is used as an example to illustrate the effectiveness of the suggested method. Rational homoclinic wave solution, a new family of two-wave solution, is obtained by inclined periodic homoclinic breather wave solution and is just a rogue wave solution. This result shows that rogue wave originates by the extreme behaviour of homoclinic breather wave in (1+1)-dimensional nonlinear wave fields.     

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.     

Discrete breathers in deformed graphene

The linear and nonlinear dynamics of elastically deformed graphene have been studied. The region of the stability of a planar graphene sheet has been represented in the space of the two-dimensional strain (? _xx , ? _yy ) with the x and y axes oriented in the zigzag and armchair directions, respectively. It has been shown that the gap in the phonon spectrum appears in graphene under uniaxial deformation in the zigzag or armchair direction, while the gap is not formed under a hydrostatic load. It has been found that graphene deformed uniaxially in the zigzag direction supports the existence of spatially localized nonlinear modes in the form of discrete breathers, the frequency of which decreases with an increase in the amplitude. This indicates soft nonlinearity in the system. It is unusual that discrete breather has frequency within the phonon spectrum of graphene. This is explained by the fact that the oscillation of the discrete breather is polarized in the plane of the graphene sheet, while the phonon spectral band where the discrete breather frequency is located contains phonons oscillating out of plane. The stability of the discrete breather with respect to the small out-of-plane perturbation of the graphene sheet has been demonstrated.     

Focusing of nonlinear wave groups in deep water

The freak wave phenomenon in the ocean is explained by the nonlinear dynamics of phase-modulated wave trains. It is shown that the preliminary quadratic phase modulation of wave packets leads to a significant amplification of the usual modulation (Benjamin-Feir) instability. Physically, the phase modulation of water waves may be due to a variable wind in storm areas. The well-known breather solutions of the cubic Schrödinger equation appear on the final stage of the nonlinear dynamics of wave packets when the phase modulation becomes more uniform.     

Phase-modulated soliton of self-induced transparency

A theory of optical self-induced transparency for a two-component soliton in the presence of phase modulation is developed. It is shown that, under these conditions, it is possible to generate a vector soliton with characteristic parameters of oscillations with the sum and difference frequencies in the region of the carrier wave frequency. Explicit analytical expressions for nonlinear wave parameters under experimentally implemented conditions are given. In particular case, a well-known solution in the form of a breather is obtained.     

Enhanced optical limiting effect in fluorine-functionalized graphene oxide

Nonlinear optical absorption of fluorine-functionalized graphene oxide (F-GO) solution was researched by the open-aperture Z-scan method using 1064 and 532 nm lasers as the excitation sources. The F-GO dispersion exhibited strong optical limiting property and the fitted results demonstrated that the optical limiting behavior was the result of a two-photon absorption process. For F-GO nanosheets, the two-photon absorption coefficients at 1064 nm excitation are 20% larger than the values at 532 nm excitation and four times larger than that of pure GO nanosheets. It indicates that the doping of fluorine can effectively improve the nonlinear optical property of GO especially in infrared waveband, and fluorine-functionalized graphene oxide is an excellent nonlinear absorption material in infrared waveband.     

Emission from,WKB quantization,and stochastic decay of sine-Gordon solitons in external fields

Various dynamical problems for solitons (kinks and breathers) of the sine-Gordon equation perturbed by small terms describing constant or time-dependent (both periodic and random) external fields in several physical problems, e.g. long Josephson junctions and weak ferromagnets, are formulated, and results, with only few details of calculations, are presented. Among these problems are energy emission from a kink or small-amplitude breather oscillating in a constant or time-periodic external field, perturbation-induced corrections to the WKB spectrum of a quantized small-amplitude breather, and stochastic decay of a weakly bound breather into a kink-antikink pair under the action of an external field.     

Breather Solutions in Periodic Media

For nonlinear wave equations existence proofs for breathers are very rare. In the spatially homogeneous case up to rescaling the sine-Gordon equation \({\partial^2_t u = \partial^2_x u - \sin (u)}\) is the only nonlinear wave equation which is known to possess breather solutions. For nonlinear wave equations in periodic media no examples of breather solutions have been known so far. Using spatial dynamics, center manifold theory and bifurcation theory for periodic systems we construct for the first time such time periodic solutions of finite energy for a nonlinear wave equation

$ s(x) \partial^2_t u(x,t) = \partial^2_x u(x,t) - q(x) u(x,t)+ r(x)u(x,t)^3, $

with spatially periodic coefficients s, q, and r on the real axis. Such breather solutions play an important role in theoretical scenarios where photonic crystals are used as optical storage.

     

双光子光学多稳态理论

本文系统地研究了双光子共振作用的光学多稳态理论。在平均场近似下求出的状态方程包含了动态斯塔克效应及弛豫时间参数。指出了在平均场近似下,F-P腔与单向环形腔的状态方程有基本相同的形式。分析了双光子光学多稳态的两种物理机构——克尔效应与饱和吸收效应。在一定条件下这两种效应的共同作用可导致双稳态、双双稳态及三稳态。进一步还给出了区分双稳态、双双稳态及三稳态的判别式。最后还研究了双光子四波混频位相复共轭反射率的双稳态及多稳态。 关键词：     

Nonequilibrium states of the soft mode anharmonic phonons

A theory of the nonequilibrium state of the soft mode optical phonons in ferroelectrics exposed to the pumping electromagnetic field has been developed. The self-consistent field equation, which determines the phase transition temperature shift and the dielectric permeability change due to a deviation of the system from the equilibrium state is derived. A two-temperature model describing the energy transport in the presence of pumping is suggested. Analytic solutions of this model both for evolving in time and for stationary states are obtained. Spatial oscillations of the soft mode temperature are predicted for the observation time approximately equal to the energy relaxation time. It is shown that nonlinear effects can be a reason of forming of the bottleneck in the energy relaxation process. An exactly solvable model of a breather: the breather with the one-site non-linearity has been constructed and investigated.     

Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons

We consider a damped, parametrically driven discrete nonlinear Klein–Gordon equation, that models coupled pendula and micromechanical arrays, among others. To study the equation, one usually uses a small-amplitude wave ansatz, that reduces the equation into a discrete nonlinear Schrödinger equation with damping and parametric drive. Here, we justify the approximation by looking for the error bound with the method of energy estimates. Furthermore, we prove the local and global existence of solutions to the discrete nonlinear Schrödinger equation. To illustrate the main results, we consider numerical simulations showing the dynamics of errors made by the discrete nonlinear equation. We consider two types of initial conditions, with one of them being a discrete soliton of the nonlinear Schrödinger equation, that is expectedly approximate discrete breathers of the nonlinear Klein–Gordon equation.     

增益色散及双光子吸收所支持的小幅孤波

通过连续波背景上的小幅近似，给出了光纤中增益色散及双光子吸收所支持的小幅暗孤波解，讨论了它们在参数空间中不同区域的演化特征，结果表明：在一些区域存在稳定的小振幅暗孤波，但在另一些区域，这种孤波是不稳定的。     

Nonlinear effects in damping of a standing wave in a gas

The problem of constructing of a uniform asymptotic approximation to the solution of the Boltzmann equation is solved for an attenuating small-amplitude standing wave in a stationary homogeneous gas in the limiting case of small Knudsen numbers ɛ. Using the multiscale technique, a regular approximation valid for time intervals up to ɛ⁻¹ is obtained. It is shown that nonlinearity of the kinetic equation leads to violation of monochromaticity of the initial perturbation and changes the damping mode.     

Nonlinear propagation of vector extremely short pulses in a medium of symmetric and asymmetric molecules

The nonlinear propagation of extremely short electromagnetic pulses in a medium of symmetric and asymmetric molecules placed in static magnetic and electric fields is theoretically studied. Asymmetric molecules differ in that they have nonzero permanent dipole moments in stationary quantum states. A system of wave equations is derived for the ordinary and extraordinary components of pulses. It is shown that this system can be reduced in some cases to a system of coupled Ostrovsky equations and to the equation intagrable by the method for an inverse scattering transformation, including the vector version of the Ostrovsky–Vakhnenko equation. Different types of solutions of this system are considered. Only solutions representing the superposition of periodic solutions are single-valued, whereas soliton and breather solutions are multivalued.     

Kuznetsov–Ma soliton and Akhmediev breather of higher-order nonlinear Schrdinger equation

In terms of Darboux transformation, we have exactly solved the higher-order nonlinear Schrdinger equation that describes the propagation of ultrashort optical pulses in optical fibers. We discuss the modulation instability(MI) process in detail and find that the higher-order term has no effect on the MI condition. Under different conditions, we obtain Kuznetsov–Ma soliton and Akhmediev breather solutions of higher-order nonlinear Schrdinger equation. The former describes the propagation of a bright pulse on a continuous wave background in the presence of higher-order effects and the soliton's peak position is shifted owing to the presence of a nonvanishing background, while the latter implies the modulation instability process that can be used in practice to produce a train of ultrashort optical soliton pulses.     

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.