Nonlinear trans-resonant waves,vortices and patterns: From microresonators to the early Universe

Perturbed wave equations are considered. Approximate general solutions of these equations are constructed, which describe wave phenomena in different physical and chemical systems. Analogies between surface waves, nonlinear and atom optics, field theories and acoustics of the early Universe can be seen in the similarities between the general solutions that govern each system. With the help of the general solutions and boundary conditions and/or resonant conditions we have derived the basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. Then, approximate analytic resonant solutions are constructed, which describe the trans-resonant transformation of harmonic waves into traveling shock-, jet-, or mushroom-like waves. The mushroom-like waves can evolve into cloud-like and vortex-like structures. The motion and oscillations of these waves and structures can be very complex. Under parametric excitation these waves can vary their velocity, stop, and change the direction of their motion. Different dynamic patterns are yielded by these resonant traveling waves in the x-t and x-y planes. They simulate many patterns observed in liquid layers, optical systems, superconductors, Bose-Einstein condensates, micro- and electron resonators. The harmonic excitation may be compressed and transformed inside the resonant band into traveling or standing particle-like waves. The area of application of these solutions and results may possibly vary from the generation of nuclear particles, acoustical turbulence, and catastrophic seismic waves to the formation of galaxies and the Universe. In particular, the formation of galaxies and galaxy clusters may be connected with nonlinear and resonant phenomena in the early Universe. (c) 2001 American Institute of Physics.     

Time reversing solitary waves

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore, we present numerical experiments for the time-reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a new Boussinesq model, both in transmission and in reflection.     

Excitation and sensing of multiple vibrating traveling waves in one-dimensional structures

Lightly damped vibrating structures normally exhibit vibration patterns that are a combination of standing waves, i.e. mode shapes. Traveling waves, on the other hand, occur only under special circumstances. In this work, the theoretical conditions under which traveling waves prevail in finite structure are investigated. These conditions are highly sensitive to the geometrical and material parameters of the structure and in particular the vibration pattern is sensitive to the boundary conditions. There are several combinations under which traveling waves cannot be formed and these ill-posed cases are analyzed in some detail. To overcome the unavoidable uncertainties in a model, a tuning process based on identification and optimization of the excitation is suggested. The identification process uses a parametric algorithm to estimate the wavenumbers of the measured vibrations. Then, the waves are decomposed into traveling and standing parts and the external excitation is tuned until a pure traveling wave is formed.     

Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.     

Nonlinear wave interactions in porous media filled with a quantum fluid

The problem of the nonlinear interaction between the fourth sound and an acoustic wave propagating in a porous medium filled with superfluid helium is solved. Based on the Landau equations of quantum fluid dynamics and on the Biot theory of mechanical waves in a porous medium, nonlinear wave equations are derived for studying the aforementioned interaction. An expression is obtained for the vertex that determines the excitation of an acoustic wave by two waves of the fourth sound. The possibility of an experimental observation of this process is estimated.     

Particle Simulation of Electrostatic Waves Driven by an Electron Beam

A one-dimensional electrostatic particle-in-cell simulation is performed to study electrostatic wave excitation due to an electron beam in a plasma system. The excited fundamental and harmonic waves are analyzed with the fast Fourier transformation and the wavelet transformation. The second harmonic is suggested to be generated by wave-wave coupling during the nonlinear evolution, which involves forward propagating and backward propagating Langmuir waves. Furthermore, the background electrons may be heated and accelerated by the electrostatic waves.     

Non-dispersive traveling waves in inclined shallow water channels

Existence of traveling waves propagating without internal reflection in inclined water channels of arbitrary slope is demonstrated. It is shown that traveling non-monochromatic waves exist in both linear and nonlinear shallow water theories in the case of a uniformly inclined channel with a parabolic cross-section. The properties of these waves are studied. It is shown that linear traveling waves should have a sign-variable shape. The amplitude of linear traveling waves in a channel satisfies the same Green's law, which is usually derived from the energy flux conservation for smoothly inhomogeneous media. Amplitudes of nonlinear traveling waves deviate from the linear Green's law, and the behavior of positive and negative amplitudes are different. Negative amplitude grows faster than positive amplitude in shallow water. The phase of nonlinear waves (travel time) is described well by the linear WKB approach. It is shown that nonlinear traveling waves of any amplitude always break near the shoreline if the boundary condition of the full absorption is applied.     

Parametric interaction of magnetostatic waves with a nonstationary local pump

A solution is obtained for the general problem of the nonstationary interaction of backward volume magnetostatic waves in films of yttrium-iron garnet with local parametric pumping. In the case of a large pump region, l≫λ, where λ is the wavelength of the backward volume magnetostatic waves, the problem reduces to a system of truncated equations for two packets of counter propagating waves. In the opposite case, l<λ, the exact problem of parametric interactions of the eigenmodes of a ferrite film (both counterpropagating and in the same direction) is solved numerically. Both cases are studied experimentally and good qualitative and quantitative agreement is obtained with the theory. For the first time, the reversal of a wave front and the time reversal of the shape of backward volume magnetostatic wave pulses are observed and a change in the propagation time for the peak of the signal pulse and a reduction in its width owing to pumping are recorded. Two operating regimes are identified for a nonstationary parametric backward volume magnetostatic wave amplifier with local pumping, which differ in the ratio of the duration of the pump pulse to the transit time for the wave through the local pump region, and the effect of the parametric excitation of two-dimensional spin waves on the interaction of backward volume magnetostatic waves with a local nonstationary parametric pump is determined. Zh. éksp. Teor. Fiz. 116, 2192–2211 (December 1999)     

Special solutions for nonlinear wave-particle interaction

A class of exact solutions for a coupled set of nonlinear equations describing the interaction between two propagating waves and a system of particles is found. These solutions include traveling wave solutions of the non-linear coupled equations.     

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.     

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrödinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrödinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrödinger equation. The stability analysis is discussed both analytically and numerically.     

Superheterodyne Amplification of Sub Millimeter Electromagnetic Waves in an n-GaAs Film

The superheterodyne amplification of sub-millimeter electromagnetic waves in GasAs due to negative differential mobility is analyzed. The nonlinearity arises from the current and the magnetic field of the electromagnetic waves. The case of interaction of two traveling counter propagating electromagnetic waves and the following space charge wave in an n-GaAs film, placed onto i-GaAs substrate, is considered, under a 2D electron gas model. The simulation of this nonlinear interaction shows a certain amplification of the sub-millimeter electromagnetic wave.     

Parametric Mechanism of Anomalous Microwave Power Absorption in Experiments on Electron Cyclotron Heating in Toroidal Magnetic Traps

The nonlinear stage of the parametric decay instability of an extraordinary wave is analyzed in the presence of a nonmonotonic density profile. The decay excites an electron Bernstein wave, which is localized in the vicinity of a local density maximum, and an ion Bernstein wave, which leaves a nonlinear interaction region and is absorbed by ions in the vicinity of the harmonics of the ion cyclotron frequency. The main mechanism of instability saturation is considered to be a cascade of decays of a primary daughter electron Bernstein wave, which leads to the excitation of localized secondary electron Bernstein waves and ion cyclotron (Bernstein) waves. The localization of electron Bernstein waves causes a significant decrease in the secondary- decay excitation threshold, which is thought to provide saturation of the primary instability at the lowest level. The saturation of the primary parametric decay instability of a pump wave and the anomalous absorption of the pump power are analytically estimated. A numerical simulation is performed using the parameters that are typical of the experiments on the electron cyclotron resonance heating of plasma at the second resonance harmonic in TCV tokamak.     

On the interaction of a shear wave with a moving domain wall in a nonlinear response of the spin subsystem

A solution of the problem of parametric interaction between a plane monochromatic shear wave and a uniformly moving 180°-domain wall of a garnet-ferrite crystal is obtained in the exchangeless magnetostatic approximation by using the perturbation method under the conditions of a nonlinear response of the spin subsystem. It is shown that in a ferromagnetic resonance with magnetostatic oscillation of stray fields, the nonlinearity of the spin subsystem leads to the excitation of shear waves of triple frequency, which may have amplitudes comparable with that of the incident wave for oscillations doubly localized by a domain wall.     

Thickness-shear modes and magnetoelastic waves in a longitudinally magnetized ferromagnetic plate

Magnetoelastic (ME) waves and thickness-shear modes in the ferromagnetic plate are studied. Coupled vibrations of magnetization and shear elastic deformations excited simultaneously by a variable magnetic field propagate in two mutually perpendicular directions: parallel and normal to a surface. For parameters characteristic of isotropic ferromagnet with the sample magnetization and Zeeman field parallel to the surface, resonant frequencies of shear modes are computed and their dispersion law is examined. It is shown that the dependence of dimensional resonances frequencies on wave number k_z of ME wave propagating along saturating field direction occurs. The possibility of excitation of ME waves with different k_z explains multimode character of thickness ME resonances.     

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.     

A Particle-in-Cell Simulation Study on Harmonic Waves Excited by Electron Beams in Unmagnetized Plasmas

The excitation of harmonic waves by an electron beam is studied with electrostatic simulations.The results suggest that the harmonic waves are excited during the linear stage of the simulation and are developed in the nonlinear stage.First,the Langmuir waves(LWs)are excited by the beam electrons.Then the coupling of the forward propagating LWs and beam modes will excite the second harmonic waves.The third harmonic waves will be produced if the lower velocity side of the beam still has a positive velocity gradient.The beam velocity decreases at the same time,which provides the energy for wave excitation.We find that it is difficult to excite the harmonic waves with the increase of the thermal velocity of the beam electrons.The beam electrons will be heated after waves are excited,and then the part of the forward propagating LWs will turn into electron acoustic waves under the condition with a large enough intensity of beam electrons.Moreover,the action of ions hardly affects the formation of harmonic waves.     

On wave solutions of a weakly nonlinear and weakly dispersive two-mode wave system

Abstract

In this paper, we introduce and study rigorously a Hamiltonian structure and soliton solutions for a weakly dissipative and weakly nonlinear medium that governs two Korteweg–de vries (KdV) wave modes. The bounded solution and traveling wave solution such as cnoidal wave and solitary wave are obtained. Subsequently, the equation is numerically solved by Fourier spectral method for its two-soliton solution. These solutions may be useful to explain the nonlinear dynamics of waves for an equation supporting multi-mode weakly dispersive and nonlinear wave medium. In addition, we give an explicit explanation of the mathematics behind the soliton phenomenon for a better understanding of the equation.