On nonlinear water waves under a light wind and Landau type equations near the stability threshold

Various mechanisms of nonlinear saturation of water wave growth under the action of a light wind are discussed. The unstable wind may be saturated by nonlinear dissipation due to the energy transfer to the damping harmonics of the wave. Other nonlinear saturation mechanisms: nonlinear frequency shift, self-modulation or self-focusing of a wave packet may be effective in certain wavenumber regions. In case the wind speed is close to the critical one, an equation is derived for the complex wave amplitude. This equation describes all these nonlinear effects in near-critical systems. In the one-dimensional case this is the nonlinear Shrödinger equation with complex coefficients. Its solutions under various conditions are discussed.     

Modulation of an internal gravity wave packet in a stratified shear flow

Modulations of an internal gravity wave packet in a stratified shear flow are discussed in the weakly nonlinear and weakly dispersive context. It is shown that the modulations are described by a variable coefficient nonlinear Schrödinger equation when the modulations are confined to the direction of wave propagation. Transverse modulations couple the nonlinear Schrödinger equation to the mean flow equations. For long waves, it is shown that the modulation equations may be somewhat simplified. An Appendix describes the equations governing long wave resonance.     

含电学边界的压电层合梁的非线性弯曲波

非线性科学己成为近代科学发展的一个重要标志,特别是非线性动力学和非线性波的研究对于解决自然科学各领域中遇到的复杂现象和问题有着极其重要的意义.本文研究了含电学边界条件的压电层合梁的非线性弯曲波传播特性.首先,考虑几何非线性效应和压电耦合效应,利用哈密顿原理建立了一维无限长矩形压电层合梁弯曲波的非线性方程.其次,采用Ja...     

Nonlinear characteristics of the interaction of a turbulent boundary layer with a wavy surface

A nonlinear interaction of a turbulent boundary layer with a wavy surface of a solid body or a liquid whose level has a deviation in the form of a traveling monochromatic wave is studied. For the waviness of small curvature, a calculation procedure is proposed for the amplitude dependence of the drag coefficient and complex elasticity which characterizes the back action of the flow on the surface inflection. The analysis is based on the use of an isotropic algebraic model of turbulent viscosity and an orthogonal system of curvilinear coordinates that follow the surface inflections. The interaction between the flow and the surface wave is described within the framework of a quasi-linear model, and a two-scale, mean-flow model is used to determine the transverse structure of the flow in a smoothly expanding boundary layer. Institute of Applied Physics, Russian Academy of Sciences, Nizhnii Novgorod 603600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 72–84, November–December, 1998.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

Exact solutions,conservation laws,bifurcation of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver equation

In the present work, we observe the dynamical behavior of nonlinear and supernonlinear traveling waves for Sharma–Tasso–Olver (STO) equation. Exact solutions are derived using \({1}/{G^{^{\prime }}}\) expansion and modified Kudryashov methods. The wave transformation is used to transform STO equation into an ordinary differential equation. Combining Runge–Kutta fourth-order and Fourier spectral technique, we use a mixed scheme for the numerical study of STO equation. Since spectral methods expand the solution in trigonometric series resulting into higher-order technique and Runge–Kutta produces improved accuracy, we extract these qualities for a mixed scheme. Results so produced are presented graphically which provide a useful information about the dynamical behavior. Bifurcation behavior of nonlinear and supernonlinear traveling waves of STO equation is studied with the help of bifurcation theory of planar dynamical systems. It is observed that STO equation supports nonlinear solitary wave, periodic wave, shock wave, stable oscillatory wave and most important supernonlinear periodic wave.     

New explicit multi-symplectic scheme for nonlinear wave equation简

Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Nonlinear analysis and solitary waves for two superposed streaming electrified fluids of uniform depths with rigid boundaries

The nonlinear modulation of the interfacial waves of two superposed dielectric fluids with uniform depths and rigid horizontal boundaries, under the influence of constant normal electric fields and uniform horizontal velocities, is investigated using the multiple-time scales method. It is found that the behavior of small perturbations superimposed on traveling wave trains can be described by a nonlinear Schrödinger equation in a frame of reference moving with the group velocity. Wave-like solutions to this equation are examined, and different types of localized excitations (envelope solitary waves) are shown to exist. It is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the nonlinear Schrödinger equation may be approximated by the well-known Korteweg-de Vries equation. The speed of the solitary on the interface is seen to be reduced by the electric field. It is found that there are two critical values of the applied voltage that lead to (i) breaking up of the solitary waves, and (ii) bifurcation of solutions of the governing equations. On the other hand, the complex amplitude of standing wave trains near the marginal state is governed by a similar type of nonlinear Schrödinger equation in which the roles of time and space are interchanged. This equation, under a suitable transformation, is obtained as the Korteweg-de Vries equation with a variable coefficient. It is shown that this type of equations admit a solitary wave type of solutions with variable speed. Using the tangent hyperbolic method, it is observed that the wave speed increases as well as decreases, with the increase of electric field values, according to the chosen wavenumbers range. Finally, the nonlinear stability analysis is discussed in view of the coefficients of nonlinear Schrödinger equation to show the effects of various physical parameters, and also to recover the some limiting cases studied earlier in the literature.     

近岸非平整海底上耗散动力系统的广义波作用量守恒方程

为刻划近岸波-流-海底相互作用耗散动力系统的多种复杂作用机制，着眼于波浪对近岸大尺度变化环境流作用和考虑多变海底地形(可典型地刻划为由慢变水深和快变水深构成)的影响，由基于黏性流体Navie-Stokes方程的平均流方程，建立了近岸耗散动力系统的广义波作用量守恒方程，从中提出垂向速度波作用量和耗散波作用量这两种新概念，使得它们和经典的波作用量相互间达成了一种互补、协调而又主次分明的更为广泛的守恒形式．从而把波作用量这一经典概念从理想的平均流守恒系统引申到实际的平均流耗散系统(即广义守恒系统)中去，为解释沿岸过程和应用于近海、海岸工程提供了一个理论基础．     

Investigation of Anisotropy-Resolving Turbulence Models by Reference to Highly-Resolved LES Data for Separated Flow

The predictive properties of several non-linear eddy-viscosity models are investigated by reference to highly-resolved LES data obtained by the authors for an internal flow featuring massive separation from a curved surface. The test geometry is a periodic segment of a channel constricted by two-dimensional (2D) `hills' on the lower wall. The mean-flow Reynolds number is 21560. Periodic boundary conditions are applied in the streamwise and spanwise directions. This makes the statistical properties of the simulated flow genuinely 2D and independent from boundary conditions, except at the walls. The simulation was performed on a high-quality, 5M-node grid. The focus of the study is on the exploitation of the LES data for the mean-flow, Reynolds stresses and macro-length-scale. Model solutions are first compared with the LES data, and selected models are then subjected to a-priori studies designed to elucidate the role of specific model fragments in the non-linear stress-strain/vorticity relation and their contribution to observed defects in the mean-flow and turbulence fields. The role of the equation governing the length-scale, via different surrogate variables, is also investigated. It is shown that, while most non-linear models overestimate the separation region, due mainly to model defects that result in insufficient shear stress in the separated shear layer, model forms can be derived which provide a satisfactory representation of the flow. One such model is identified. This combines a particular quadratic constitutive relation with a wall-anisotropy term, a high-normal-strain correction and a new form of the equation for the specific dissipation ω = ∈/k. This revised version was published online in July 2006 with corrections to the Cover Date.     

Singular Limits of the Klein–Gordon Equation

We establish the singular limits, including semiclassical, nonrelativistic and nonrelativistic-semiclassical limits, of the Cauchy problem for the modulated defocusing nonlinear Klein–Gordon equation. For the semiclassical limit, ${\hbar\to 0}$ , we show that the limit wave function of the modulated defocusing cubic nonlinear Klein–Gordon equation solves the relativistic wave map and the associated phase function satisfies a linear relativistic wave equation. The nonrelativistic limit, c → ∞, of the modulated defocusing nonlinear Klein–Gordon equation is the defocusing nonlinear Schrödinger equation. The nonrelativistic-semiclassical limit, ${\hbar\to 0, c=\hbar^{-\alpha}\to \infty}$ for some α > 0, of the modulated defocusing cubic nonlinear Klein–Gordon equation is the classical wave map for the limit wave function and a typical linear wave equation for the associated phase function.     

L'équation de la stropholyseThe stropholytic equation

It is shown that the modelling of homogeneous turbulent flows with mean rotation requires the consideration of stropholytic effects, besides componentality and dimensionality effects. Stropholytic effects are directly related to the mean-flow rotationality. The equation for the purely symmetric stropholysis is established and prerequisites for the closure of terms involved in this equation are discussed. To cite this article: J. Piquet, C. R. Mecanique 331 (2003).     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.     

Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinear irrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and the number of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steep waves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximum modulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundred time periods. We have also carried out some much longer computations, over several thousands of time periods in which several steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulations using weakly nonlinear theory, which leads to the nonlinear Schrödinger equation. The steeper events are shorter in both space and time than the lower events. Solutions of the nonlinear Schrödinger equation can be transformed from one steepness to another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellent agreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomes an almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinear Schrödinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep wave event. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs.     

Modelling a recirculating density-driven turbulent flow

A mathematical model of turbulent density-driven flows is presented and is solved numerically. A form of the k–? turbulence model is used to characterize the turbulent transport, and both this non-linear model and a sediment transport equation are coupled with the mean-flow fluid motion equations. A partitioned, Newton–Raphson-based solution scheme is used to effect a solution. The model is applied to the study of flow through a circular secondary sedimentation basin.     

Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.