一维非线性声波传播特性

针对一维非线性声波的传播问题进行了有限元仿真和实验研究. 首先推导了一维非线性声波方程的有限元形式, 含有高阶矩阵的非线性项导致声波具有波形畸变、谐波滋生、基频信号能量向高次谐波传递等非线性特性. 编制有限元程序对一维非线性声波进行了计算并对仿真得到的畸变非线性声波信号进行处理, 分析其传播性质和物理意义. 为验证有限元计算结果, 开展了水中的非线性声波传播的实验研究, 得到了不同输入信号幅度激励下和不同传播距离的畸变非线性声波信号. 然后对基波和二次谐波的传播性质进行详细讨论, 分析了二次谐波幅度与传播距离和输入信号幅度的变化关系及其意义, 拟合出二次谐波幅度随传播距离变化的方程并阐述了拟合方程的物理意义. 结果表明, 数值仿真信号及其频谱均与实验结果有较好的一致性, 证实计算方法和结果的正确性, 并提出了具有一定物理意义的二次谐波随传播距离变化的简单数学关系. 最后还对固体中的非线性声波传播性质进行了初步探讨. 本研究工作可为流体介质中的非线性声传播问题提供理论和实验依据.     

Analysis of the rogue waves in the blood based on the high-order NLS equations with variable coefficients

The research of rogue waves is an advanced field which has important practical and theoretical significances in mathematics, physics, biological fluid mechanics, oceanography, etc. Using the reductive perturbation theory and long wave approximation, the equations governing the movement of blood vessel walls and the flow of blood are transformed into high-order nonlinear Schrödinger (NLS) equations with variable coefficients. The third-order nonlinear Schrödinger equation is degenerated into a completely integrable Sasa-Satsuma equation (SSE) whose solutions can be used to approximately simulate the real rogue waves in the vessels. For the first time, we discuss the conditions for generating rogue waves in the blood vessels and effects of some physiological parameters on the rogue waves. Based on the traveling wave solutions of the fourth-order nonlinear Schrödinger equation, we analyze the effects of the higher order terms and the initial deformations of the blood vessel on the wave propagation and the displacement of the tube wall. Our results reveal that the amplitude of the rogue waves are proportional to the initial stretching ratio of the tube. The high-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave steepness.     

Large variations in NLS bi-soliton wave groups

The nonlinear Schrödinger (NLS) equation describes the spatial–temporal evolution of the complex amplitude of wave groups in beams and pulses in both second and third order nonlinear material. In this paper we investigate in detail the wave group that has the exact two-soliton solution as amplitude, and show that large variations in the amplitude appear to form a pattern that, at the peak interaction, resembles quite well the linear superposition. The complexity of the phenomenon is a combination of nonlinear effects and linear interference of the carrier waves: the characteristic parameter is the quotient of wave amplitude and frequency difference of the carrier waves, which is also proportional to the quotient of the modulation period of the carrier waves during interaction and the interaction period of the soliton envelopes.     

A high-order nonlinear envelope equation for gravity waves in finite-depth water

A third-order nonlinear envelope equation is derived for surface waves in finite-depth water by assuming small wave steepness, narrow-band spectrum, and small depth as compared to the modulation length. A generalized Dysthe equation is derived for waves in relatively deep water. In the shallow-water limit, one of the nonlinear dispersive terms vanishes. This limit case is compared with the envelope equation for waves described by the Korteweg-de Vries equation. The critical regime of vanishing nonlinearity in the classical nonlinear Schrödinger equation for water waves (when kh ≈ 1.363) is analyzed. It is shown that the modulational instability threshold shifts toward the shallow-water (long-wavelength) limit with increasing wave intensity.     

Third-order Stokes wave solutions for interfacial internalwaves in three-layer dendity-stratified fluid

<正>Interfacial internal waves in a three-layer density-stratified fluid are investigated using a singular perturbation method,and third-order asymptotic solutions of the velocity potentials and third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory.As expected,the third-order solutions describe the third-order nonlinear modification and the third-order nonlinear interactions between the interfacial waves.The wave velocity depends on not only the wave number and the depth of each layer but also on the wave amplitude.     

Acoustic shock wave propagation in a heterogeneous medium: a numerical simulation beyond the parabolic approximation

Numerical simulation of nonlinear acoustics and shock waves in a weakly heterogeneous and lossless medium is considered. The wave equation is formulated so as to separate homogeneous diffraction, heterogeneous effects, and nonlinearities. A numerical method called heterogeneous one-way approximation for resolution of diffraction (HOWARD) is developed, that solves the homogeneous part of the equation in the spectral domain (both in time and space) through a one-way approximation neglecting backscattering. A second-order parabolic approximation is performed but only on the small, heterogeneous part. So the resulting equation is more precise than the usual standard or wide-angle parabolic approximation. It has the same dispersion equation as the exact wave equation for all forward propagating waves, including evanescent waves. Finally, nonlinear terms are treated through an analytical, shock-fitting method. Several validation tests are performed through comparisons with analytical solutions in the linear case and outputs of the standard or wide-angle parabolic approximation in the nonlinear case. Numerical convergence tests and physical analysis are finally performed in the fully heterogeneous and nonlinear case of shock wave focusing through an acoustical lens.     

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.     

Short optical solitons in fibers

The dynamics of short (of the order of a few wave periods) intense optical pulses and interaction of short optical solitons in fibers are considered within the framework of the third-order nonlinear Schrodinger equation. It is shown that an initial pulse tends to one or a few short solitons plus a linear quasiperiodic wave when the third-order linear dispersion and nonlinear dispersion have parameters of the same sign. The number and parameters of the solitons depend on the magnitudes of initial pulse parameters. Interaction of short optical solitons having different amplitudes is accompanied by radiation of part of the wave field from the area of interaction, by an increase of the soliton with larger amplitude, and a decrease of the soliton with a smaller one. (c) 2000 American Institute of Physics.     

Dispersion and nonlinear frequancy shift of natural waves in the Bose–Einstein condensate

Quantum mechanics equations for a system of the Bose particles are represented in the form of material field equations. A nonlinear equation for the macroscopic one-particle wave function is derived. Using the Krylov–Bogolyubov–Mitropol’skii method for equations in partial derivatives, nonlinear waves in the Bose–Einstein condensate are investigated. In the cubic approximation, dispersion relations for waves are derived and nonlinear frequency shift is calculated in the first- and third-order approximations for the interaction radius.     

Terahertz-wave generation in a conventional optical fiber

We theoretically propose surface-emitted and collinear phase-matched terahertz (THz)-wave generation in a conventional optical fiber. The third-order nonlinear effect, four-wave mixing (FWM), is used to generate THz waves in an optical fiber. Surface-emitted THz-wave generation via FWM is realized using a single-mode fiber. Perfect phase matching is obtained at ~800 nm and 1.5 microm pumping, and it follows that third-order polarization in an optical fiber has the same phase at any point. In this situation, the optical fiber acts like a phased array antenna of the THz wave. Collinear phase-matching THz waves are obtained under the same conditions as for surface-emitted THz waves, and the THz wave is propagated in the silica cladding of the optical fiber. This is a promising method for realizing a reasonable THz-wave source.     

Rogue Wave Solutions for Nonlinear Schrödinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrödinger (NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.     

Nichtlineare Wechselwirkung elektromagnetischer Wellen im Plasma mit konstantem Magnetfeld

The nonlinear interaction of waves in anisotropic plasmas is considered. Solving the hydrodynamics equations with a n-th order perturbation procedure, an equation for the nonlinear electric field is derived. The n-th order current produced by interaction of waves with lower order is represented by means of a nonlinear tensor of conductivity, which is calculated for n = 2. It is shown, that a miced plasma wave with combination frequencies is excited as the result of interaction of two transverse (ordinary) waves propagating perpendicular to the magnetic field.     

Numerical and physical modeling of the tomography process based on third-order nonlinear acoustic effects

The possibility of employing the nonlinear effect of generation of third-order combination waves for the purposes of medical diagnostics is analyzed. This effect can be used to reconstruct the spatial distribution of acoustic nonlinear parameters in the framework of the wave approach. Contributions of third-order nonlinear scattering itself and of the double second-order scattering are evaluated. These two competing processes evolve simultaneously and produce similar observed effects, which can nevertheless be separated. A two-dimensional experimental scheme that contains only three transmitters and one receiver, uses two primary wideband modulated waves and an introduced third monochromatic wave, is proposed. Results of the numerical and physical model experiments are provided.     

Cumulative solutions of nonlinear longitudinal vibration in isotropic solid bars

Based on the strain invariant relationship and taking the high-order elastic energy into account, a nonlinear wave equation is derived, in which the excitation, linear damping, and the other nonlinear terms are regarded as the first-order correction to the linear wave equation. To solve the equation, the biggest challenge is that the secular terms exist not only in the fundamental wave equation but also in the harmonic wave equation （unlike the Duffing oscillator, where they exist only in the fundamental wave equation）. In order to overcome this difficulty and to obtain a steady periodic solution by the perturbation technique, the following procedures are taken： （i） for the fundamental wave equation, the secular term is eliminated and therefore a frequency response equation is obtained; （ii） for the harmonics, the cumulative solutions are sought by the Lagrange variation parameter method. It is shown by the results obtained that the second- and higher-order harmonic waves exist in a vibrating bar, of which the amplitude increases linearly with the distance from the source when its length is much more than the wavelength; the shift of the resonant peak and the amplitudes of the harmonic waves depend closely on nonlinear coefficients; there are similarities to a certain extent among the amplitudes of the odd- （or even-） order harmonics, based on which the nonlinear coefficients can be determined by varying the strain and measuring the amplitudes of the harmonic waves in different locations.     

Short Envelope Solitons. Isotropic and Anisotropic Media

Within the framework of the third-order approximation of the nonlinear wave dispersion theory, we find new classes of short scalar and vector solitons of lengths about several wavelengths. Short scalar solitons are found within the framework of a third-order nonlinear Schrödinger equation (NSE-3) including both the nonlinear dispersion terms and the third-order linear dispersion term. The interaction of such solitons is studied, and the soliton stability is proved. Short vector solitons are found within the framework of a coupled third-order nonlinear Schröodinger equation (CNSE-3). Interaction and stability of such solitons are studied.     

Dynamics of wave packets and soliton interaction in terms of the third-order nonlinear Schrödinger equation

We present the results of numerical study of the evolution of wave packets and envelope soliton interaction in terms of the third-order nonlinear Schrödinger equation. It is shown that an arbitrary initial pulse evolves to a few solitons and a linear quasiperiodic wave. The interaction of solitons is accompanied by the radiation of part of the wave field in the form of a linear quasiperiodic wave from the interaction region, amplification of the soliton with larger amplitude and attenuation of the soliton with smaller amplitude.     

The scattering of ultrasound by cylinders: implications for diffraction tomography

The validity of wave equations employed as system models in acoustical diffraction tomography is investigated using simulations and measurements of the scattering of plane ultrasound waves by cylinders. It is demonstrated by simulation and experiment that it can be appropriate to neglect density fluctuations and shear waves, implying that the commonly used form of the wave equation suitably describes scattering by fluctuations of acoustic speed and absorption. Diffraction tomographic reconstructions of simulated data reveal the importance of absorption, the behavior of the real and imaginary parts of the reconstructed refractive index, and the relative advantages and limitations of the Born and Rytov approximate transformations.     

On Modulational Interaction of the Lower-Hybrid Drift Oscillations

The general nonlinear equation of the third order in field strength for the lower-hybrid drift waves in inhomogeneous plasma is obtained on the basis of kinetic theory. This equation enables us to describe strong turbulence effects (modulational instability, soliton-like solutions, etc.) as well as weak turbulence effects (decays, scattering). The investigation of the modulational instability of the lower-hybrid drift waves is carried out. It is demonstrated that the development of the lower-hybrid drift wave modulational instability is possible only when the wavevector of the modulational perturbations is less or of the order of the wavevector of the pump wave. The condition on the wave vectors, when the nonlinear response defining the character of the modulational instability is determined by the inhomogeneity effects, is obtained.