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.     

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.     

Growth of wave penetration under wind amplification

The penetration of nonlinear wind waves into the depth of a liquid is investigated in a laboratory channel. It is found that the amplitude of nonlinear waves on deep water depends only on the wave steepness normalized by the wave length. An expression is found that connects the growth of the wave mass with the wave steepness.     

Intensification of three-dimensional waves by wind on deep water

It was found that the intensification of waves by wind includes consecutive cycles of growth of the longest three-dimensional wave’s steepness and its decay into even longer waves. The ratio of the lengths of waves that arise during decay was obtained as a function of the Froude number, and the steepness of the leading slope of the longest wave was obtained as a function of the wave’s steepness. An expression describing the depth of nonlinear waves’ penetration into a water column as a function of their steepness was found.     

Energy spectrum of the ensemble of weakly nonlinear gravity-capillary waves on a fluid surface

We consider nonlinear gravity-capillary waves with the nonlinearity parameter ? ～ 0.1–0.25. For this nonlinearity, time scale separation does not occur and the kinetic wave equation does not hold. An energy cascade in this case is built at the dynamic time scale (D-cascade) and is computed by the increment chain equation method first introduced in [15]. We for the first time compute an analytic expression for the energy spectrum of nonlinear gravity-capillary waves as an explicit function of the ratio of surface tension to the gravity acceleration. We show that its two limits—pure capillary and pure gravity waves on a fluid surface—coincide with the previously obtained results. We also discuss relations of the D-cascade model with a few known models used in the theory of nonlinear waves such as Zakharov’s equation, resonance of modes with nonlinear Stokes-corrected frequencies, and the Benjamin-Feir index. These connections are crucial in understanding and forecasting specifics of the energy transport in a variety of multicomponent wave dynamics, from oceanography to optics, from plasma physics to acoustics.     

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.     

On the modulation instability of surface waves on a large-scale shear flow

The problem of the modulation instability of a weakly nonlinear quasi-monochromatic wave on the surface of deep water in the presence of a steady-state collinear large-scale inhomogeneous flow (e.g., of a jet type) has been considered. In a certain range of (obtuse) angles of incidence of the wave with respect to the flow direction, the steepness of the refracted wave increases considerably, which contributes to the enhancement of nonlinear effects, including the formation of so-called rogue waves. The corresponding nonlinear Schrödinger equation with variable coefficients suitable for the analysis of the modulation instability has been derived.     

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.     

From bell-shaped solitary wave to W/M-shaped solitary wave solutions in an integrable nonlinear wave equation

The bifurcation theory of dynamical systems is applied to an integrable nonlinear wave equation. As a result, it is pointed out that the solitary waves of this equation evolve from bell-shaped solitary waves to W/M-shaped solitary waves when wave speed passes certain critical wave speed. Under different parameter conditions, all exact explicit parametric representations of solitary wave solutions are obtained.     

Nonlinear stage of the Benjamin-Feir instability: three-dimensional coherent structures and rogue waves

A specific, genuinely three-dimensional mechanism of rogue wave formation, in a late stage of the modulational instability of a perturbed Stokes deep-water wave, is recognized through numerical experiments. The simulations are based on fully nonlinear equations describing weakly three-dimensional potential flows of an ideal fluid with a free surface in terms of conformal variables. Spontaneous formation of zigzag patterns for wave amplitude is observed in a nonlinear stage of the instability. If initial wave steepness is sufficiently high (ka>0.06), these coherent structures produce rogue waves. The most tall waves appear in turns of the zigzags. For ka<0.06, the structures decay typically without formation of steep waves.     

Nonlinear fast growth of water waves under wind forcing

In the wind-driven wave regime, the Miles mechanism gives an estimate of the growth rate of the waves under the effect of wind. We consider the case where this growth rate, normalised with respect to the frequency of the carrier wave, is of the order of the wave steepness. Using the method of multiple scales, we calculate the terms which appear in the nonlinear Schrödinger (NLS) equation in this regime of fast-growing waves. We define a coordinate transformation which maps the forced NLS equation into the standard NLS with constant coefficients, that has a number of known analytical soliton solutions. Among these solutions, the Peregrine and the Akhmediev solitons show an enhancement of both their lifetime and maximum amplitude which is in qualitative agreement with the results of tank experiments and numerical simulations of dispersive focusing under the action of wind.     

Nonlinear and linear wave phenomena in narrow pipes

Phenomena arising in the course of wave propagation in narrow pipes are considered. For nonlinear waves described by the generalized Webster equation, a simplified nonlinear equation is obtained that allows for low-frequency geometric dispersion causing an asymmetric distortion of the periodic wave profile, which qualitatively resembles the distortion of a nonlinear wave in a diffracted beam. Tunneling of a wave through a pipe constriction is investigated. Possible applications of the phenomenon are discussed, and its relation to the problems of quantum mechanics because of the similarity of the basic equations of the Klein-Gordon and Schrödinger types is pointed out. The importance of studying the tunneling of nonlinear waves and broadband signals is indicated.     

Variation of wave action: Modulations of the phase shift for strongly nonlinear dispersive waves with weak dissipation

Strongly nonlinear dispersive waves described by a general Klein—Gordon equation with slowly varying coefficients and a dissipative perturbation are analyzed using the method of multiple scales. We use the exact equation of wave action. The spatial and temporal slow modulations of the phase shift are shown to be governed by a new equation, which results from linearization of the wave action, its flux, and its dissipation due to perturbations of the slow parameters: frequency and wave number (vector). This result extends to nonlinear partial differential equations, the quite recent work by the authors on nonlinear oscillations governed by ordinary differential equations.     

Rogue Waves of the Higher-Order Dispersive Nonlinear Schrödinger Equation

In this paper,the rogue waves of the higher-order dispersive nonlinear Schrödinger (HDNLS) equation are investigated,which describes the propagation of ultrashort optical pulse in optical fibers.The rogue wave solutions of HDNLS equation are constructed by using the modified Darboux transformation method.The explicit first and second-order rogue wave solutions are presented under the plane wave seeding solution background.The nonlinear dynamics and properties of rogue waves are discussed by analyzing the obtained rational solutions.The influence of little perturbation ε on the rogue waves is discussed with the help of graphical simulation.     

Application of higher-order KdV——mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere

Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV--mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.     

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.     

A(2+1)-dimensional nonlinear model for Rossby waves in stratified fluids and its solitary solution

In this paper,we investigate a(2+1)-dimensional nonlinear equation model for Rossby waves in stratified fluids.We derive a forced Zakharov–Kuznetsov(ZK)–Burgers equation from the quasigeostrophic potential vorticity equation with dissipation and topography under the generalized beta effect,and by utilizing temporal and spatial multiple scale transform and the perturbation expansion method.Through the analysis of this model,it is found that the generalized beta effect and basic topography can induce nonlinear waves,and slowly varying topography is an external impact factor for Rossby waves.Additionally,the conservation laws for the mass and energy of solitary waves are analyzed.Eventually,the solitary wave solutions of the forced ZK–Burgers equation are obtained by the simplest equation method as well as the new modified ansatz method.Based on the solitary wave solutions obtained,we discuss the effects of dissipation and slowly varying topography on Rossby solitary waves.     

Anomalous wave as a result of the collision of two wave groups on the sea surface

The numerical simulation of the nonlinear dynamics of the sea surface has shown that the collision of two groups of relatively low waves with close but noncollinear wave vectors (two or three waves in each group with a steepness of about 0.2) can result in the appearance of an individual anomalous wave whose height is noticeably larger than that in the linear theory. Since such collisions quite often occur on the ocean surface, this scenario of the formation of rogue waves is apparently most typical under natural conditions.     

Density waves in traffic flow model with relative velocity

The car-following model of traffic flow is extended to take into account the relative velocity. The stability condition of this model is obtained by using linear stability theory. It is shown that the stability of uniform traffic flow is improved by considering the relative velocity. From nonlinear analysis, it is shown that three different density waves, that is, the triangular shock wave, soliton wave and kink-antikink wave, appear in the stable, metastable and unstable regions of traffic flow respectively. The three different density waves are described by the nonlinear wave equations: the Burgers equation, Korteweg-de Vries (KdV) equation and modified Korteweg-de Vries (mKdV) equation, respectively.     

Two-Mode Wave Solutions to the Degasperis--Procesi Equation

By introducing a new type of solutions, called the multiple-mode wave solutions which can be expressed in nonlinear superposition of single-mode waves with different speeds, we investigate the two-mode wave solutions in Degasperis-Procesi equation and two cases are derived. The explicit expressions for the two-mode waves as well as the existence conditions are presented. It is shown that the two-mode waves may be the nonlinear combinations of many types of single-mode waves, such as periodic waves, solJtons, compactons, etc., and more complicated multiple-mode waves can be obtained if higher order or more single-mode waves are taken into consideration. It is pointed out that the two-mode wave solutions can be employed to display the typical mechanism of the interactions between different single-mode waves.