Experimental and numerical study of the propagation of focused wave groups in the nearshore zone

The propagation of focused wave groups in intermediate water depth and the shoaling zone is experimentally and numerically considered in this paper. The experiments are carried out in a two-dimensional wave flume and wave trains derived from Pierson-Moskowitz and JONSWAP spectrum are generated. The peak frequency does not change during the wave train propagation for Pierson-Moskowitz waves; however, a downshift of this peak is observed for JONSWAP waves. An energy partitioning is performed in order to track the spatial evolution of energy. Four energy regions are defined for each spectrum type. A nonlinear energy transfer between different spectral regions as the wave train propagates is demonstrated and quantified. Numerical simulations are conducted using a modified Boussinesq model for long waves in shallow waters of varying depth. Experimental results are in satisfactory agreement with numerical predictions, especially in the case of wave trains derived from JONSWAP spectrum.     

非线性管理光纤光栅中的调制不稳定性研究

基于非线性耦合模方程,研究了非线性管理光纤光栅中的调制不稳定性,得到了调制不稳定性的色散关系。与常规的非线性光纤光栅相比,在非线性管理光纤光栅中,克尔非线性的变化改变了调制不稳定性增益谱的谱宽和幅度,并导致新的不稳定性区域的出现:在反常色散区,原来关于零波数对称的两个旁瓣随着克尔非线性变化的增加其增益幅度递减至零,在经历了一段无增益区域之后,又逐渐形成了在零波数附近区域的一个新的单峰;而在正常色散区,除了原来的两个增益区域之外,零波数附近出现了新的增益区,增益的幅度随克尔非线性变化的增加而递增。可见,非线性管理光纤光栅给调制不稳定性的产生提供了更多的空间。     

正压大气环流中的曲面周期波和孤波

大尺度正压大气环流的波动特征对理解气候变化具有重要的意义,而非线性浅水波方程组是描述大尺度正压大气环流的原始控制方程.本文对线性方程的复变函数解,通过二次适当的移植,求得浅水波方程组的发展方程的扰动位势的实变函数解,该实变函数解析解由基流项和波动项两部分组成.其中基流由波数、波速、β效应、变形半径和时间的任意函数共同决定;波动项与β效应有关.分析表明,在大尺度正压大气环流中扰动位势存在曲面的周期波和孤波的现象,周期波与孤波相互调制而呈现不稳定性;当多个周期孤波同时出现时,则彼此独立传播;扰动位势波动项中的时间任意函数对曲面周期孤波的波幅有调制作用,可控制波的产生、发展和消失.所得结果对研究大气波动现象和气候变化具有一定的理论参考价值.     

五阶非线性光纤中连续谱相位扰动下的光传输与脉冲串产生

根据包含五阶非线性的扩展非线性薛定谔方程,数值研究了高斯型连续谱相位扰动而不是传统单色扰动下基于调制不稳定性的高重复率脉冲串产生.结果表明:脉冲串也能像传统情形那样形成,但却呈现出不同的特性.如脉冲数目有限,且各脉冲的高度、强度及间距不等.脉冲数目随传输距离增加而增加.而五阶非线性能使脉冲宽度和间距变小因而有利于高重复率脉冲串产生,负五阶非线性则相反.对脉冲串形成过程中演变啁啾的数值计算表明,啁啾及其随距离的变化都是高度非单调的,五阶非线性将改变啁啾的范围和量值.     

Rogue waves: Classification,measurement and data analysis,and hyperfast numerical modeling

The last twenty years has seen the birth and subsequent evolution of a fundamental new idea in nonlinear wave research: Rogue waves, freak waves or extreme events in the wave field dynamics can often be classified as coherent structure solutions of the requisite nonlinear partial differential wave equations (PDEs). Since a large number of generic nonlinear PDEs occur across many branches of physics, the approach is widely applicable to many fields including the dynamics of ocean surface waves, internal waves, plasma waves, acoustic waves, nonlinear optics, solid state physics, geophysical fluid dynamics and turbulence (vortex dynamics and nonlinear waves), just to name a few. The first goal of this paper is to give a classification scheme for solutions of this type using the inverse scattering transform (IST) with periodic boundary conditions. In this context the methods of algebraic geometry give the solutions of particular PDEs in terms of Riemann theta functions. In the classification scheme the Riemann spectrum fully defines the coherent structure solutions and their mutual nonlinear interactions. I discuss three methods for determining the Riemann spectrum: (1) algebraic-geometric loop integrals, (2) Schottky uniformization and (3) the Nakamura-Boyd approach. I give an overview of several nonlinear wave equations and graph some of their coherent structure solutions using theta functions. The second goal is to discuss how theta functions can be used for developing data analysis (nonlinear Fourier) algorithms; nonlinear filtering techniques allow for the extraction of coherent structures from time series. The third goal is to address hyperfast numerical models of nonlinear wave equations (which are thousands of times faster than traditional spectral methods).     

Nonlinear Fourier analysis of localized wave fields

Summary The Fourier transform (FT) has long served as an indispensable means for analysing wave motion described by linear evlution equations. The methods are well known and include not only mathematical analysis but also the analysis of data as well. In recent years new spectral methods have been developed for analysing nonlinear evolution equations. Such methods are generalizations of the FT to specific nonlinear wave systems and are referred to as the spectral or scattering transform (ST). Herein we use numerical procedures for applying the ST directly to the analysis of localized data described by the Kortewegde Vries (KdV) equation on the infinite interval,i.e. the Cauchy problem in shallow water. In this context we emphasize the importance of the direct spectral transform (DST) as a wave number domain representation of nonlinear data. The numerical methods discussed for the KdV equation should be extendible to the large class of systems considered by Ablowitzet al., Calogero and Degasperis. We give examples of the spectral analysis of nonlinear, computer-generated data.

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Riassunto La ben nota trasformata di Fourier (FT) è di uso commune per analizzare moti ondosi descritti da equazioni di evoluzione lineari. Negli ultimi anni sono stati sviluppati nouvi metodi (detti trasformate spettrali o di scattering, ST) per l'analisi di equazioni di evoluzione non lineari basati sulla generalizzazione della FT agli specifici sistemi ondosi non lineari. Qui usiamo procedimenti numerici per applicare la ST direttamente all'analisi di dati localizzati descritti dall'equazione di Korteweg-deVries nell'intervallo infinito, cioè al problema di Cauchy in acqua bassa. In questo contesto sottolineiamo l'importanza della trasformata specttrale diretta come rappresentazione nel campo dei numeri d'onda di dati non lineari. I metodi numerici discussi, che a titolo di esempio sono qui applicati all'analisi spettrale di dati non lineari generati dal calcolatore, hanno l'interessante possibilità di essere estendibili ai problemi piú generali di propagazione ondosa.

     

Study of diffusion of wave packets in a square lattice under external fields along the discrete nonlinear Schrödinger equation

The object of the present work is to analyze the effect of nonlinearity on wave packet propagation in a square lattice subject to a magnetic and an electric field in the Hall configuration, by using the Discrete Nonlinear Schrödinger Equation (DNLSE). In previous works we have shown that without the nonlinear term, the presence of the magnetic field induces the formation of vortices that remain stationary, while a wave packet is introduced in the system. As for the effect of an applied electric field, it was shown that the vortices propagate in a direction perpendicular to the electric field, similar behavior as presented in the classical treatment, we provide a quantum mechanics explanation for that. We have performed the calculations considering first the action of the magnetic field as well as the nonlinearity. The results indicate that for low values of the nonlinear parameter U the vortices remain stationary while preserving the form. For greater values of the parameter the picture gets distorted, the more so, the greater the nonlinearity. As for the inclusion of the electric field, we note that for small U, the wave packet propagates perpendicular to the applied field, until for greater values of U the wave gets partially localized in a definite region of the lattice. That is, for strong nonlinearity the wave packet gets partially trapped, while the tail of it can propagate through the lattice. Note that this tail propagation is responsible for the over-diffusion for long times of the wave packet under the action of an electric field. We have produced short films that show clearly the time evolution of the wave packet, which can add to the understanding of the dynamics.     

非线性强度任意二聚的非线性链的透射性质

用离散的非线性薛定谔的递推关系研究了非线性强度任意二聚的非线性链的透射性质. 结果表明该链存在一个共振透射态,共振态的能量为非线性强度与入射波振幅模平方的乘积; 取出射波振幅为定值和取入射波振幅模为定值来计算透射系数,其结果在非共振态有明显的差别: 取出射波振幅为定值时电子的透射随能量为单值函数,而取入射波振幅模为定值时电子的透射呈现多稳态. 并指出只有取入射波振幅模为定值时才能真正反映非线性强度对电子透射性质的影响.     

The nonlinear Schrödinger equation and the propagation of weakly nonlinear waves in optical fibers and on the water surface

The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. a nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.     

Nonlinear waves in Group III metals

The effect of nonlinearity on the possibility of Group III metals to support propagating radio waves is investigated theoretically. A nonlinear wave is shown to be able to propagate in a metal when the magnetic field of a wave of a large amplitude traps holes and causes collisionless cyclotron absorption to be suppressed.     

Evolution of basic equations for nearshore wave field

In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications.     

On a new class of <Emphasis Type="Italic">S</Emphasis>-polarized surface polaritons at the interface between two media with saturated nonlinearity

It has been shown that S-polarized surface polaritons can propagate along the interface between two media with saturated nonlinearity. The profile of the wave amplitude has a maximum in one of the contacting media. An expression has been obtained for the wave field amplitude at the interface. The relation between the surface wave vector and wave frequency can be arbitrary.     

An Intense Wave in Defective Media Containing Both Quadratic and Modular Nonlinearities: Shock Waves,Harmonics, and Nondestructive Testing

The observed nonclassical power-law dependence of the amplitude of the second harmonic wave on the amplitude of a harmonic pump wave is explained as a phenomenon associated with two types of nonlinearity in a structurally inhomogeneous medium. An approach to solving the inverse problem of determining the nonlinearity parameters and the exponent in the above-mentioned dependence is demonstrated. To describe the effects of strongly pronounced nonlinearity, equations containing a double nonlinearity and generalizing the Hopf and Burgers equations are proposed. The possibility of their exact linearization is demonstrated. The profiles, spectral composition, and average wave intensity in such doubly nonlinear media are calculated. The shape of the shock front is found, and its width is estimated. The wave energy losses that depend on both nonlinearity parameters—quadratic and modular—are calculated.     

New approaches to nonlinear diffractive field propagation

In many domains of acoustic field propagation, such as medical ultrasound imaging, lithotripsy shock treatment, and underwater sonar, a realistic calculation of beam patterns requires treatment of the effects of diffraction from finite sources. Also, the mechanisms of loss and nonlinear effects within the medium are typically nonnegligible. The combination of diffraction, attenuation, and nonlinear effects has been treated by a number of formulations and numerical techniques. A novel model that incrementally propagates the field of baffled planar sources with substeps that account for the physics of diffraction, attenuation, and nonlinearity is presented. The model accounts for the effect of refraction and reflection (but not multiple reflections) in the case of propagation through multiple, parallel layers of fluid medium. An implementation of the model for axis symmetric sources has been developed. In one substep of the implementation, a new discrete Hankel transform is used with spatial transform techniques to propagate the field over a short distance with diffraction and attenuation. In the other substep, the temporal frequency domain solution to Burgers' equation is implemented to account for the nonlinear accretion and depletion of harmonics. This approach yields a computationally efficient procedure for calculating beam patterns from a baffled planar, axially symmetric source under conditions ranging from quasilinear through shock. The model is not restricted by the usual parabolic wave approximation and the field's directionality is explicitly accounted for at each point. Useage of a harmonic-limiting scheme allows the model to propagate some previously intractable high-intensity nonlinear fields. Results of the model are shown to be in excellent agreement with measurements performed on the nonlinear field of an unfocused 2.25-MHz piston source, even in the near field where the established parabolic wave approximation model fails. Next, the model is used to compare the water path and in situ fields of a medical ultrasound device. Finally, the model is used to calculate the spatial heating rate associated with a nonlinear field and to simulate the phenomenon of saturation-induced beam broadening.     

非线性内波存在下的浅海远程混响模型

研究了浅海非线性内波对远程混响场的影响。通过分析非线性内波活动引起声源到海底散射元以及散射元到接收点之间的声传播变化,给出了非线性内波引起远程混响强度变化的表示,建立了非线性内波存在下的浅海远程混响模型,数值计算了非线性内波运动引起远程混响强度的变化。理论和数值计算表明,非线性内波的活动能够引起远程混响强度的变化,在某些情况下会导致远程混响强度增强。通过讨论非线性内波引声简正波的耦合效应,给出了其引起远程混响强度增强的原因。      

Modulated periodic stokes waves in deep water

Modulated deep-water 1D Stokes waves are considered experimentally and theoretically. Wave trains are modulated in a controlled fashion and their evolution is recorded. Data from repeated laboratory experiments are reproducible near the wave maker, but diverge away from the wave maker. Numerical integration of a perturbed nonlinear Schrodinger equation and an associated linear spectral problem indicate that under suitable conditions modulated periodic Stokes waves evolve chaotically. Sensitive spectral evolution in the neighborhood of homoclinic manifolds of the unperturbed nonlinear Schrodinger equation is found.     

Equilibrium distribution of the wave energy in a carbyne chain

The steady-state energy distribution of thermal vibrations at a given ambient temperature has been investigated based on a simple mathematical model that takes into account central and noncentral interactions between carbon atoms in a one-dimensional carbyne chain. The investigation has been performed using standard asymptotic methods of nonlinear dynamics in terms of the classical mechanics. In the first-order nonlinear approximation, there have been revealed resonant wave triads that are formed at a typical nonlinearity of the system under phase matching conditions. Each resonant triad consists of one longitudinal and two transverse vibration modes. In the general case, the chain is characterized by a superposition of similar resonant triplets of different spectral scales. It has been found that the energy equipartition of nonlinear stationary waves in the carbyne chain at a given temperature completely obeys the standard Rayleigh–Jeans law due to the proportional amplitude dispersion. The possibility of spontaneous formation of three-frequency envelope solitons in carbyne has been demonstrated. Heat in the form of such solitons can propagate in a chain of carbon atoms without diffusion, like localized waves.     

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.     

Observation of conical waves in focusing, dispersive, and dissipative Kerr media

Excitation of unbalanced-Bessel beams by a gradual increase of nonlinearity in a water sample outlines the achievement of the first ever observed quasimonochromatic wave packet that propagates stably for hundreds of Rayleigh lengths in a focusing and dispersive Kerr medium, i.e., in the absence of spectral broadening and conical emission. A modulational instability analysis reveals the key role of nonlinear dissipation in quenching the growth of spatiotemporal unstable modes.     

On the Effective Metric for an Electromagnetic Wave Propagating in the Field of an Intense Laser Radiation

It is shown that in the nonlinear electrodynamics of vacuum, which is a corollary of quantum electrodynamics, a weak electromagnetic wave propagates in the field of an intense laser radiation as if this took place in some effective space-time whose metric tensor depends on the electric field of the laser radiation. The components of the metric vector of the effective space-time along the geodesic lines of which the rays of the weak electromagnetic wave propagate in the given case have been found. It is pointed out that this property may serve as a basis for observing the manifestations of the nonlinearity of the electrodynamics of vacuum.