Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.     

Nonlinear evolution of a three-dimensional transverse wave packet in a hot plasma including the effect of its interaction with an ion-acoustic wave

By a perturbation method two coupled nonlinear partial differential equations are obtained for the nonlinear evolution of a three dimensional transverse wave packet in a hot plasma including the effect of its interaction with a long wavelength ion-acoustic wave. From these two equations a nonlinear dispersion relation is obtained, from which the instability condition of a uniform transverse wave train including the effect of its interaction, both at resonance and at nonresonance with a long wavelength ion-acoustic wave, are deduced. Resonance occurs when the component of group velocity of the longitudinal wave along the direction of propagation of the ion-acoustic wave is equal to the phase velocity of the wave. Assuming the usual type of dependence of amplitude on space and time the coupled equations are transformed into two other coupled equations, which reduced to a single nonliear Schrödingsr equation when three dimensionality is disregarded. It is found that these three transformed equations cannot give instability condition at resonance.On leave fromThe Department of Mathematics, University of Kalyani, West Bengal, India.     

Ultrasonic waves in crystals with charged dislocations

A system of equations for charged dislocations, where the quadratic nonlinear terms are taken into account, is derived using the variational principle. This system describes the propagation of ultrasonic (US) waves in crystals with charged dislocations. From the linearized system of equations a linear dispersion equation is derived. Formulas for the phase linear velocity of the wave and the absorption coefficient are obtained, which show essential influence of charged dislocations and electrical properties of media on the mentioned quantities. For a nonlinear US wave an equation for the amplitude of the first harmonic is derived and, as a consequence, expressions are obtained for the nonlinear velocity of the US wave, for the attenuation of the first harmonic's amplitude, and for phase variation.     

Semiclassical trajectory-coherent states of the nonlinear Schrödinger equation with unitary nonlinearity

Semiclassically concentrated states of the nonlinear Schrödinger equation (NLSE) with unitary nonlinearity, representing multidimensional localized wave packets, are constructed on the basis of the Maslov complex germ theory. A system of ordinary differential equations of Hamilton-Ehrenfest (HE) type, describing the motion of the wave packet centroid, is derived. The structure of the HE system is strongly influenced by the initial conditions of the Cauchy problem for the NLSE. Wave packets of Gaussian type are constructed in an explicit form. Possible use of the solutions constructed in the problem of optical pulse propagation in a nonlinear medium with nonstationary dispersion is discussed.     

Propagation and modulational instability of Rossby waves in stratified fluids

Perturbation analysis and scale expansion are used to derive the (2+1)-dimensional coupled nonlinear Schrödinger (CNLS) equations that can describe interactions of two Rossby waves propagating in stratified fluids. The (2+1)-dimensional equations can reflect and describe the wave propagation more intuitively and accurately. The properties of the two waves in the process of propagation can be analyzed by the solution obtained from the equations using the Hirota bilinear method, and the influence factors of modulational instability are analyzed. The results suggest that, when two Rossby waves with slightly different wave numbers propagate in the stratified fluids, the intensity of bright soliton decreases with the increases of dark soliton coefficients. In addition, the size of modulational instable area is related to the amplitude and wave number in y direction.     

Effects of initial compression stress on wave propagation in carbon nanotubes

An analytical method to investigate wave propagation in single- and double- walled carbon nanotubes under initial compression stress is presented. The nanotube structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The governing equations are derived based on Flügge equations of motion. Frequency equations of wave propagation in single and double wall carbon nanotubes are described through the effects of initial compression stress and van der Waals force. To show the effects of Initial compression stress on the wave propagation in nanotubes, the symmetrical mode can be analyzed based on the present elastic continuum model. It is shown that the wave speed are sensitive to the compression stress especially for the lower frequencies.     

具有几个光振荡周期的飞秒激光脉冲

从麦克斯韦方程组出发 ,推导出了具有几个光振荡周期的飞秒激光脉冲在非线性光纤中传输的方程和非线性光纤的折射率。给出了描述具有几个光振荡周期的飞秒激光脉冲在非线性光纤中传输方程的解。研究了在非线性光纤中自相位调制导致具有几个光振荡周期的飞秒激光脉冲频谱展宽 (脉宽压缩 )的详细物理过程。研究了非线性光纤中飞秒光孤子产生的条件     

柱面非线性麦克斯韦方程组的行波解

柱面电磁波在各种非均匀非线性介质中的传播问题具有非常重要的研究价值.对描述该问题的柱面非线性麦克斯韦方程组进行精确求解,则是最近几年新兴的研究热点.但由于非线性偏微分方程组的极端复杂性,针对任意初边值条件的精确求解在客观上具有极高的难度,已有工作仅解决了柱面电磁波在指数非线性因子的非色散介质中的传播情况.因此,针对更为确定的物理场景,寻求能够精确描述其中更为广泛的物理性质的解,是一种更为有效的处理方法.本文讨论了具有任意非线性因子与幂律非均匀因子的非色散介质中柱面麦克斯韦方程组的行波精确解,理论分析表明这种情况下柱面电磁波的电场分量E已不存在通常形如E=g(r-kt)的平面行波解;继而通过适当的变量替换与求解相应的非线性常微分方程,给出电场分量E=g(lnr-kt)形式的广义行波解,并以例子展示所得到的解中蕴含的类似于自陡效应的物理现象.     

Linear and nonlinear waves in crystals with charged dislocations

Based on the principle, a set of equations describing ultrasonic wave propagation in a dielectric crystal with charged dislocations has been derived. The linear dispersion relation has been derived, and the ultrasonic wave velocity and absorbance have been determined. In the case of the nonlinear ultrasonic wave, equations for its amplitude damping and phase variation have been obtained.     

Plasma wave frequency shift in a weak transverse magnetic field due to trapped particle acceleration

A simple model of nonlinear electrostatic wave–particle interaction in a weak magnetic field perpendicular to the direction of wave propagation is developed. The damping of the wave loaded with the phase bunched groups of trapped particles is considered with the aid of the model equations. To determine the nonlinear frequency shift of the wave in the process of the trapped particle acceleration, the nonlinear dispersion equation is derived. It is shown that the corresponding variation of the phase velocity may affect the interaction process and hence must be taken into account in the self-consistent treatment of the time evolution of the wave.     

An iterative method to solve a regularized model for strongly nonlinear long internal waves

We present a simple iterative scheme to solve numerically a regularized internal wave model describing the large amplitude motion of the interface between two layers of different densities. Compared with the original strongly nonlinear internal wave model of Miyata [10] and Choi and Camassa [2], the regularized model adopted here suppresses shear instability associated with a velocity jump across the interface, but the coupling between the upper and lower layers is more complicated so that an additional system of coupled linear equations must be solved at every time step after a set of nonlinear evolution equations are integrated in time. Therefore, an efficient numerical scheme is desirable. In our iterative scheme, the linear system is decoupled and simple linear operators with constant coefficients are required to be inverted. Through linear analysis, it is shown that the scheme converges fast with an optimum choice of iteration parameters. After demonstrating its effectiveness for a model problem, the iterative scheme is applied to solve the regularized internal wave model using a pseudo-spectral method for the propagation of a single internal solitary wave and the head-on collision between two solitary waves of different wave amplitudes.     

Solitary waves in a binary nonlinear waveguide array

Based on the coupled-mode theory, the propagation of light pulses is studied analytically for a system of an infinite number of tunnel-coupled parallel equidistant waveguides of optically nonlinear materials; in the considered system, waveguides with a positive refractive index alternate with waveguides with a negative refractive index. Partial solutions to a system of nonlinear equations describing the evolution of these pulses are found in the case in which fields in adjacent waveguides differ only in the phase factor. For a solitary wave formed by coupled wave packets localized each in its own waveguide, these solutions describe the stationary propagation in a definite direction. It is shown that the coupling strength between waveguides has an effect on the propagation rate of the obtained stationary pulses.     

Flexural and transversal wave motion in homogeneous isotropic thermoelastic plates by using asymptotic method

The present investigation is concerned with the flexural and transversal wave motion in an infinite, transversely isotropic, thermoelastic plate by asymptotic method. The governing equations for the flexural and transversal motions have been derived from the system of three-dimensional dynamical equations of linear theory of coupled thermoelasticity. The asymptotic operator plate model for free vibrations; both flexural and transversal, in a homogenous thermoelastic plate leads to fifth degree and cubic polynomial secular equations, respectively, that governs frequency and phase velocity of various possible modes of wave propagation at all wavelengths. All the coefficients of differential operator have been expressed as explicit functions of the material parameters. The velocity dispersion equations for the flexural and transversal wave motion have been deduced from the three-dimensional analog of Rayleigh-Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity have also been derived. The thermoelastic Rayleigh-Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established with that of asymptotic method. The dispersion curves for phase velocity, group velocity and attenuation coefficient of various flexural and transversal wave modes are shown graphically for aluminum-epoxy material elastic and thermoelastic plates.     

Nonlinear theory of transverse perturbations of quasi-one-dimensional solitons

An averaged variational principle is applied to analyze the nonlinear effect of transverse perturbations (including diffraction) on quasi-one-dimensional soliton propagation governed by various wave equations. It is shown that parameters of the spatiotemporal solitons described by the cubic Schrödinger equation and the Yajima-Oikawa model of interaction between long-and short-wavelength waves satisfy the spatial quintic nonlinear Schrödinger equation for a complex-valued function composed of the amplitude and eikonal of the soliton. Three-dimensional solutions are found for two-component “bullets” having long-and short-wavelength components. Vortex and hole-vortex structures are found for envelope solitons and for two-component solitons in the regime of resonant long/short-wave coupling. Weakly nonlinear behavior of transverse perturbations of one-dimensional soliton solutions in a self-defocusing medium is described by the Kadomtsev-Petviashvili equation. The corresponding rationally localized “lump” solutions can be considered as secondary solitons propagating along the phase fronts of the primary solitons. This conclusion holds for primary solitons described by a broad class of nonlinear wave equations.     

Dynamics of one- and two-dimensional kinks in bistable reaction-diffusion equations with quasidiscrete sources of reaction

We study the evolution of fronts in a bistable reaction-diffusion system when the nonlinear reaction term is spatially inhomogeneous. This equation has been used to model wave propagation in various biological systems. Extending previous works on homogeneous reaction terms, we derive asymptotically an equation governing the front motion, which is strongly nonlinear and, for the two-dimensional case, generalizes the classical mean curvature flow equation. We study the motion of one- and two-dimensional fronts, finding that the inhomogeneity acts as a "potential function" for the motion of the front; i.e., there is wave propagation failure and the steady state solution depends on the structure of the function describing the inhomogeneity. (c) 2001 American Institute of Physics.     

Generalized thermoelastic waves in homogeneous isotropic plates

The propagation of thermoelastic waves in homogeneous isotropic plate subjected to stress-free and rigid insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT), Lord-Shulman (LS), Green-Lindsay (GL), and Green-Nagdhi (GN) theories of thermoelasticity. Secular equations for the plate in closed form and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. It is shown that the motion for SH modes gets decoupled from the rest of the motion and remains unaffected due to thermo-mechanical coupling and thermal relaxation effects. The phase velocities for SH modes have also been obtained. The results for coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. At short wavelength limits the secular equations for symmetric and skew-symmetric waves in a stress-free insulated and isothermal plate reduce to Rayleigh surface waves frequency equations. Finally, the numerical solution is carried out for aluminum-epoxy composite material and the dispersion curves for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results.     

Nonlinear dynamics of the interface between fluids at the suppression of Kelvin–Helmholtz instability by a tangential electric field

The nonlinear dynamics of the interface between ideal dielectric fluids in the presence of tangential discontinuity of the velocity at the interface and the stabilizing action of the horizontal electric field is examined. It is shown that the regime of motion of the interface where liquids move along the field lines occurs in the state of neutral equilibrium where electrostatic forces suppress Kelvin–Helmholtz instability. The equations of motion of the interface describing this regime can be reduced to an arbitrary number of ordinary differential equations describing the propagation and interaction of structurally stable solitary waves, viz. rational solitons. It is shown that weakly interacting solitary waves recover their shape and velocity after collision, whereas strongly interacting solitary waves can form a wave packet (breather).     

Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory

The governing equation of wave motion of viscoelastic SWCNTs (single-walled carbon nanotubes) with surface effect under magnetic field is formulated on the basis of the nonlocal strain gradient theory. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that the size-dependent effects on the phase velocity may be ignored at low wave numbers, however, is significant at high wave numbers. Phase velocity can increase by decreasing damping or increasing the intensity of magnetic field. The damping ratio considering surface effect is larger than that without considering surface effect. Damping ratio can increase by increasing damping, increasing wave number, or decreasing the intensity of magnetic field.     

Study on the propagation of coupling shock waves with phase transition under combined tension-torsion impact loading

Phase transition can strongly change the stress wave propagation features. In this paper, the characteristic wave propagation under combined tension and torsion impact loading was studied with a simplified constitutive model of phase transition considering both pressure and shear stress. The results showed that for loading from the austenitic phase to the mixed phase, the wave propagation was similar to that in the elasto-plastic materials. However, for an instantaneous loading from the austenitic phase or mixed phase directly to the martensitic phase, a coupling shock wave (CSHW) with phase transition was predicted due to the second phase strengthening effect, which has barely been studied before. Through analysis of the constitutive equations with phase transition and the discontinuity conditions of shock waves, the control equations of the generalized Hugoniot curve was obtained and the CSHW problem with phase transition was solved analytically. An independent numerical simulation of step loading along a NiTi thin walled tube suffering a combined tension-torsion impact loading was given to prove the existence of CSHW. The simulation discloses the formation mechanism of CSHW and the adjusting process of the stress state ahead of CSHW, which reflects the intrinsic characteristic of materials with strong nonlinear constitutive behavior.     

行波管中慢电磁行波与电子注非线性互作用普遍理论

在同时考虑多信号输入和相对论效应的情况下，利用波导激励理论获得了行波管中慢电磁行波与电子注非线性互作用的全三维自洽工作方程组，包括激发方程、运动方程、能量转化方程、相位演化方程等，适合大部分行波管中慢电磁行波与电子注的非线性互作用过程.利用该理论具体分析了一个宽带螺旋线行波管在多信号输入时的交叉调制，并与实验结果进行了比较，验证了理论和计算的正确性.另外，还模拟了一个相对论盘荷波导行波管中的非线性注波互作用过程. 关键词： 行波管 慢电磁行波 非线性注波互作用 交叉调制