暗怪波的相互作用

以耦合非线性薛定谔方程为理论模型,数值研究了两个一阶暗怪波在正常色散单模光纤中的相互作用.基于一阶暗怪波精确解,采用分步傅里叶数值模拟法,从间距、相位差和振幅系数比方面讨论相邻两个一阶暗怪波之间的相互作用.基于二阶暗怪波精确解,讨论了两个一阶暗怪波的非线性相互作用.研究结果表明:同相位情况下,间距参数T1为0、5、20时,相邻两个一阶暗怪波相互作用激发产生“扭结型”暗怪波.相比较于单个暗怪波发生能量的弥散,“扭结型”暗怪波分裂形成多个次暗怪波.反相位情况下,间距参数T1为2、7、12时,相邻两个一阶暗怪波相互作用也可以激发产生“扭结型”暗怪波.并且“扭结型”暗怪波初始激发的空间位置偏离原始单个暗怪波的位置5.振幅系数比越大,该空间位置越接近5.二阶暗怪波可以看作是两个一阶暗怪波的非线性叠加,复合型和三组分型二阶暗怪波与相邻两个一阶暗怪波的相互作用略有相似.     

有流存在时三层流体界面波的二阶Stokes波解

以小振幅波理论为基础，利用摄动方法研究了有背景流场存在时密度三层成层状态下的界面内波，得到了各层流体速度势的二阶渐近解及界面内波波面位移的二阶Stokes波解，并讨论了界面波的Kelvin-Helmholtz不稳定性.结果表明：有流存在的情况下三层密度成层流体界面内波的一阶渐近解(线性波解)、频散关系及二阶渐近解不仅依赖于各层流体的厚度和密度，也依赖于各层流体的背景流场；界面内波波面位移的二阶Stokes波解不仅描述了界面波之间的二阶非线性相互作用，也描述了背景流与界面波之间的二阶非线性相互作用；当每层流 关键词： 界面波 均匀流 二阶Stokes波解 Kelvin-Helmholtz不稳定性     

A shock-capturing methodology based on adaptative spatial filtering for high-order non-linear computations

A shock-capturing methodology is developed for non-linear computations using low-dissipation schemes and centered finite differences. It consists in applying an adaptative second-order filtering to handle discontinuities in combination with a background selective filtering to remove grid-to-grid oscillations. The shock-capturing filtering is written in its conservative form, and its magnitude is determined dynamically from the flow solutions. A shock-detection procedure based on a Jameson-like shock sensor is derived so as to apply the shock-capturing filtering only around shocks. A second-order filter with reduced errors in the Fourier space with respect to the standard second-order filter is also designed. Linear and non-linear 1D and 2D problems are solved to show that the methodology is capable of capturing shocks without providing dissipation outside shocks. The shock detection allows in particular to distinguish shocks from linear waves, and from vortices when it is performed from dilatation rather than from pressure. Finally the methodology is simple to implement and reasonable in terms of computational cost.     

各向异性固体板中的非线性超声导波

将超声导波的非线性视为线性波动响应的一个二阶微扰,采用二阶微扰近似和导波的模式展开分析方法,从理论上研究了各向异性固体板中超声导波的二次谐波发生效应,给出了超声导波二次谐波声场的一般形式解。板材的各向异性可从多个方面对二次谐波发生效应产生影响,尤其是板材各向异性引起的导波模式相速度的改变,可显著地影响到构成导波二次谐波声场的二倍频导波模式随传播距离的积累增长程度。根据理论分析和数值计算结果可知,在不同传播方向上超声导波的非线性效应存在较显著的差异,伴随基频导波模式的传播所发生的二次谐波对固体板材的各向异性表现出更为敏感的性质,这一结果可为准确定征板材的弹性各向异性提供一种理论依据。     

三层流体界面内波的二阶Stokes解

以小振幅波理论为基础，利用摄动方法研究了三层密度成层状态下的界面内波，求得了三层成层状态下各层速度势的二阶渐近解及界面内波波面位移的二阶Stokes解.结果表明：一阶解为正弦波解，与传统线性理论的结果相一致；二阶解描述了界面波的二阶非线性修正及两界面波之间的非线性相互作用；一阶解及二阶解都依赖于各层流体的厚度及密度.Umeyama导出的理论结果为本文的特殊情形. 关键词： 三层密度成层流体 内波 二阶Stokes解 小振幅波理论     

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从理论上研究了固体板内SH型板波的\     

Numerical wave propagation on non-uniform one-dimensional staggered grids

The wave propagation behaviour of centered difference schemes on one-dimensional non-uniform staggered grids is investigated. Previous results for the linear advection equation are extended to the case of the shallow water equations on staggered grids. For waves of a given frequency, the wave field is decomposed into right- and left-propagating components, and a wave energy conservation law is derived in terms of these components. For slowly varying grids, separate evolution equations for the right- and left-propagating components are derived, leading to the result that there is asymptotically no reflection in the limit of a slowly varying grid, provided that waves of that frequency are resolvable. However, there will be reflection from any location at which the wave group velocity goes to zero. The possibility for wave energy to tunnel through a narrow region of the grid too coarse for propagation is noted. Grids with an abrupt jump in resolution are also investigated. It is possible to tailor the scheme at the jump to minimize spurious wave reflection over a range of frequencies provided the waves are resolvable on both sides of the jump. However, it does not appear possible to avoid complete reflection, except by introducing extra dissipation terms, if the waves are not resolvable on one side of the jump. An example is presented of a second-order accurate scheme that spontaneously radiates waves from the resolution jump.     

Localized waves of the coupled cubic–quintic nonlinear Schrdinger equations in nonlinear optics

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schr o¨dinger(CCQNLS) equations through the generalized Darboux transformation(DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first-and second-order semi-rational localized waves including several free parameters are mainly discussed:(i) the semi-rational solutions degenerate to the first-and second-order vector rogue wave solutions;(ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons;(iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α.These results further uncover some striking dynamic structures in the CCQNLS system.     

三层密度分层流体毛细重力波二阶Stokes波解

以小振幅波理论为基础,利用摄动方法研究了三层密度分层流体的毛细重力波,给出了三层成层状态下各层流体速度势的二阶渐近解及毛细重力波波面位移的二阶Stokes波解.结果表明:一阶解及二阶解除了依赖于各层流体的厚度及密度,与表面张力也有很重要的关系.     

Second-order random wave solutions for interfacial internal waves in N-layer density-stratified fluid

This paper studies the random internal wave equations describing the density interface displacements and the velocity potentials of N-layer stratified fluid contained between two rigid walls at the top and bottom. The density interface displacements and the velocity potentials were solved to the second-order by an expansion approach used by Longuet-Higgins （1963） and Dean （1979） in the study of random surface waves and by Song （2004） in the study of second- order random wave solutions for internal waves in a two-layer fluid. The obtained results indicate that the first-order solutions are a linear superposition of many wave components with different amplitudes, wave numbers and frequencies, and that the amplitudes of first-order wave components with the same wave numbers and frequencies between the adjacent density interfaces are modulated by each other. They also show that the second-order solutions consist of two parts： the first one is the first-order solutions, and the second one is the solutions of the second-order asymptotic equations, which describe the second-order nonlinear modification and the second-order wave-wave interactions not only among the wave components on same density interfaces but also among the wave components between the adjacent density interfaces. Both the first-order and second-order solutions depend on the density and depth of each layer. It is also deduced that the results of the present work include those derived by Song （2004） for second-order random wave solutions for internal waves in a two-layer fluid as a particular case.     

A possibility of compensating the effect of self-action on the efficiency of second harmonic generation by femtosecond pulses

The feasibility of compensating the self-action of waves involved in second harmonic generation (SHG) by femtosecond pulses in the case of group matching is shown. Propagation of pulses is described in terms of the Schrödinger equation with the inclusion of the second-order dispersion and cubic nonlinearity. Analytical and numerical studies are performed indicating that, with some constraints imposed on the cubic nonlinearity, the SHG process occurs in the same way as in the absence of nonlinearity within a rather long propagation path.     

Rogue wave solutions for a generalized nonautonomous nonlinear equation in a nonlinear inhomogeneous fiber

In this paper, we investigate a generalized nonautonomous nonlinear equation which describes the ultrashort optical pulse propagating in a nonlinear inhomogeneous fiber. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions for the generalized nonautonomous nonlinear equation are obtained, under some variable–coefficient constraints. Properties of the first- and second-order rogue waves are graphically presented and analyzed: When the coefficients are all chosen as the constants, we can observe the some functions, the shapes of wave crests and troughs for the first- and second-order rogue waves change. Oscillating behaviors of the first- and second-order rogue waves are observed when the coefficients are the trigonometric functions.     

Second-order solutions for random interfacial waves in N-layer density-stratified fluid with steady uniform currents

In the present paper, the random interfacial waves in N-layer density-stratified fluids moving at different steady uniform speeds are researched by using an expansion technique, and the second-order asymptotic solutions of the random displacements of the density interfaces and the associated velocity potentials in N-layer fluid are presented based on the small amplitude wave theory. The obtained results indicate that the wave-wave second-order nonlinear interactions of the wave components and the second-order nonlinear interactions between the waves and currents are described. As expected, the solutions include those derived by Chen （2006） as a special case where the steady uniform currents of the N-layer fluids are taken as zero, and the solutions also reduce to those obtained by Song （2005） for second-order solutions for random interfacial waves with steady uniform currents if N = 2.     

NONLINEAR ACOUSTICS IN HIGHER-ORDER APPROXIMATION

Based on Lighthill's equation, the n-th order (n > 2) inhomogeneous wave equations are established by means of the perturbation method. The third-order nonlinear parameter C/A is defined as a new characteristic parameter in addition to the second-order one B/A, By using the Lagrange's parameter variation method, the third-order harmonic waves are obtained, in which the accumulation solution have a term proportional to the square of the distance, It is shown by analysis that all the solutions are valid only in the region where the accumulation terms are predominant.     

Approximate Homotopy Direct Reduction Method: Infinite Series Reductions to Perturbed mKdV Equations

An approximate homotopy direct reduction method is proposed and applied to two perturbed modified Korteweg- de Vries （mKdV） equations with fourth-order dispersion and second-order dissipation. The similarity reduction equations are derived to arbitrary orders. The method is valid not only for single soliton solutions but also for the Painlevd Ⅱ waves and periodic waves expressed by Jacobi elliptic functions for both fourth-order dispersion and second-order dissipation. The method is also valid for strong perturbations.     

Second-Order Resonant Interaction of Ring Current Protons with Whistler-Mode Waves

We present a study on the second-order resonant interaction between the ring current protons with Whistler-mode waves propagating near the quasi electrostatic limit following the previous second-order resonant theory. The diffusion coefficients are proportional to the electric field amplitude E, much greater than those for the regular first-order resonance, which are proportional to the electric field amplitudes square E^2. Numerical calculations for the pitch angle scattering are performed for typical energies of protons Ek = 50 keV and 100 keV at locations L = 2 and L = 3.5. The timescale for the loss process of protons by the Whistler waves is found to approach one hour, comparable to that by the EMIC waves, suggesting that Whistler waves may also contribute significantly to the ring current decay under appropriate conditions.     

Elastohydrodynamic problems in quasicrystal elasticity theory and wave propagation

Elastohydrodynamic problems of decagonal quasicrystals are analysed where the phonon field obeys wave equation and the phason field obeys diffusive wave equation. Basic equations are solved in the quasiperiodic plane and periodic plane, respectively. Final governing equations of dynamic behaviours of decagonal quasicrystals are obtained. A general solution is derived in terms of introduced three auxiliary functions, where two individually satisfy a fourth-order partial differential equation and one satisfies a second-order hyperbolic diffusion equation. Using the derived governing equations, elastic waves propagating in the quasiperiodic plane and a plane containing the period axis are analysed. Secular equations are obtained. It is found that differing from conventional crystals, at least four branches of elastic waves exist when the phonon–phason coupling is present. Moreover, acoustic waves have attenuation during wave propagation. Phason fluctuations exhibit exponential decaying behaviour due to kinematic viscosity and damping. The phase speeds are isotropic in the quasiperiodic plane and anisotropic in a plane with the periodic axis. The section of the slowness surfaces is plotted.     

Long wave-short wave resonance in nonlinear negative refractive index media

We show that long wave-short wave resonance can be achieved in a second-order nonlinear negative refractive index medium when the short wave lies on the negative index branch. With the medium exhibiting a second-order nonlinear susceptibility, a number of nonlinear phenomena such as solitary waves, paired solitons, and periodic wave trains are possible or enhanced through the cascaded second-order effect. Potential applications include the generation of terahertz waves from optical pulses.     

A standing wave model for acoustic pumping effect in microchannels

An acoustic impedance pump is comprised of a compressible section coupled at both ends to sections of different acoustic impedances. Liquid can be pumped from one end to another if the compressible section is actuated at certain locations. This paper presents an analytical model on the acoustic pumping effect in microchannels. A one-dimensional wave equation is developed for acoustic pressures in the compressible section, taking into account the actuations as acoustic source terms. The solution for the acoustic pressure is a set of standing waves established inside the compressible section, corresponding to the actuations. The pumping effect is attributed to the second-order terms of the acoustic pressures. Two control parameters are identified. One is the resonance frequency associated with the sound wave speed and length of the compressible section, and the other is the damping factor. The analytical results are compared with the experimental data, and a qualitative agreement is observed in terms of frequency characteristics of the pumping pressure.     

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrodinger system in an inhomogeneous optical fiber

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schr?dinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion(GVD) and fourth-order dispersion(FOD) coefficients are the constants, we exhibit the first-and second-order vector semirational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first-and second-order periodic vector semi-rational rogue waves, first-and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.