Exact Solitary Water Waves with Vorticity

The solitary water wave problem is to find steady free surface waves which approach a constant level of depth in the far field. The main result is the existence of a family of exact solitary waves of small amplitude for an arbitrary vorticity. Each solution has a supercritical parameter value and decays exponentially at infinity. The proof is based on a generalized implicit function theorem of the Nash–Moser type. The first approximation to the surface profile is given by the “KdV” equation. With a supercritical value of the surface tension coefficient, a family of small amplitude solitary waves of depression with subcritical parameter values is constructed for an arbitrary vorticity.     

海沟内海底地震激发的表面波动

通过底面运动学边界条件引入底面运动影响,采用高阶Boussinesq方程计算了光滑海底变形引起的表面波动形态.对于线性问题,与线性势流波浪理论进行了比较,二者结果符合良好.运用高阶Boussinesq波浪模型,针对冲绳海沟的实际地形,模拟海沟内不同震级的海底地震激发的海啸,分析了不同强度地震引起的表面波扰动形态及其非线性和色散效应.     

Asymptotic Linear Stability of Solitary Water Waves

In this paper, we study the well-posedness of Cahn–Hilliard equations with degenerate phase-dependent diffusion mobility. We consider a popular form of the equations which has been used in phase field simulations of phase separation and microstructure evolution in binary systems. We define a notion of weak solutions for the nonlinear equation. The existence of such solutions is obtained by considering the limits of Cahn–Hilliard equations with non-degenerate mobilities.     

Establishment of Surface Waves Generated by Hydrodynamic Singularities in Plane Flow

A method of direct proof of the existence of steady-state regimes in classical problems of small waves on the surface of a plane flow of ideal heavy fluid of infinite depth accompanying the uniform motion of a hydrodynamic singularity inside the flow or the operation of a pulsating immersed source is proposed.     

Solitary Waves in a Collisionless Plasma with Isothermal Pressure

Wave resonances in the hydrodynamic model of an isotropic collisionless quasi-neutral hot plasma with isothermal ions and electrons are considered. These resonances lead to the formation of two types of solitary waves: solitary waves proper and generalized solitary waves. The latter result from the nonlinear resonance of the proper solitary waves with magnetosonic and Alfvén periodic waves. The possibility of observing these waves in the Earth's magnetospheric plasma is discussed.     

On the Existence and Conditional Energetic Stability of Solitary Gravity-Capillary Surface Waves on Deep Water

This paper presents an existence and stability theory for gravity-capillary solitary waves on the surface of a body of water of infinite depth. Exploiting a classical variational principle, we prove the existence of a minimiser of the wave energy E{{\mathcal E}} subject to the constraint I=?2m{{\mathcal I}=\sqrt{2}\mu}, where I{{\mathcal I}} is the wave momentum and 0 < m << 1{0 < \mu \ll 1} . Since E{{\mathcal E}} and I{{\mathcal I}} are both conserved quantities a standard argument asserts the stability of the set D _μ of minimisers: solutions starting near D _μ remain close to D _μ in a suitably defined energy space over their interval of existence. In the applied mathematics literature solitary water waves of the present kind are modelled as solutions of the nonlinear Schr?dinger equation with cubic focussing nonlinearity. We show that the waves detected by our variational method converge (after an appropriate rescaling) to solutions of this model equation as mˉ 0{\mu \downarrow 0} .     

On the Amplitude Equations for Weakly Nonlinear Surface Waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter (Contemp Math, vol 100, pp 185–202. AMS, 1989), and more recently by Benzoni-Gavage and Rosini (Comput Math Appl 57(3–4):1463–1484, 2009), as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage (Differ Integr Equ 22(3–4):303–320, 2009) under an appropriate stability condition originally pointed out by Hunter. The same stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov (Benzoni-Gavage et al. in Adv Math 227(6):2220–2240, 2011). In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. We also show that the resulting amplitude equation has a Hamiltonian structure when the original boundary value problem has a variational origin. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity.     

Existence and Conditional Energetic Stability of Capillary-Gravity Solitary Water Waves by Minimisation

Firstly, the two-dimensional stationary water-wave problem is considered. Existence of capillary-gravity solitary waves is proved by minimising a functional related to Smales amended potential. We first establish the existence of periodic solutions of arbitrarily large periods, leading to a minimising sequence in L²() that stays away from the boundary of the neighbourhood of 0 W^2,2() in which the analysis is carried out. With the help of the concentration-compactness principle, we then show that every minimising sequence has a subsequence that, after possible shifts in the propagation direction, converges in L²() to a minimiser. Secondly, for the evolutionary problem, we prove that the set of minimal solitary waves as a whole is energetically conditionally stable. Energetically means that the distance to the set of all minimisers is defined in terms of the total energy, and conditionally means that we consider solutions to the evolutionary problem that do not explode instantaneously but could perhaps explode in finite time (e.g., via the explosion of another norm). We work in some bounded set in W^2,2() that contains the quiescent state and we are not interested in the fate of solutions that leave this set.     

Surface Water Waves and Tsunamis

Because of the enormous earthquake in Sumatra on December 26, 2004, and the devastating tsunami which followed, I have chosen the focus of my mini-course lectures at this year’s PASI to be on two topics which involve the dynamics of surface water waves. These topics are of interest to mathematicians interested in wave propagation, and particularly to Chilean scientists, I believe, because of Chile’s presence on the tectonically active Pacific Rim. My first lecture will describe the equations of fluid dynamics for the free surface above a body of fluid (the ocean surface), and the linearized equations of motion. From this, we can predict the travel time of the recent tsunami from its epicenter off of the north Sumatra coast to the coast of nearby Thailand, the easy coasts of Sri Lanka and south India, and to Africa. In fact, the signal given by ocean waves generated by the Sumatra earthquake was felt globally; within 48 h distinguishable tsunami waves were measured by wave gages in Antarctica, Chile, Rio di Janeiro, the west coast of Mexico, the east coast of the United States, and at Halifax, Nova Scotia. To describe ocean waves, we will formulate the full nonlinear fluid dynamical equations as a Hamiltonian system [19], and we will introduce the Greens function and the Dirichlet-Neumann operator for the fluid domain along with the harmonic analysis of the theory of their regularity. From an asymptotic theory of scaling transformations, we will derive the known Boussinesq-like systems and the KdV and KP equations, which govern the asymptotic behavior of tsunami waves over an idealized flat bottom. When the bottom is no longer assumed to be perfectly flat, a related theory [6, 13] gives a family of model equations taking this into account. My second lecture will describe a series of recent results in PDE, numerical results, and experimental results on the nonlinear interactions of solitary surface water waves. In contrast with the case of the KdV equations (and certain other integrable PDE), the Euler equations for a free surface do not admit clean (‘elastic’) interactions between solitary wave solutions. This has been a classical concern of oceanographers for several decades, but only recently have there been sufficiently accurate and thorough numerical simulations which quantify the degree to which solitary waves lose energy during interactions [3, 4]. It is striking that this degree of ‘inelasticity’ is remarkably small. I will describe this work, as well as recent results on the initial value problem which are very relevant to this phenomenon [14, 18].     

高度非线性孤立波与弹性大板的耦合作用研究

一维颗粒链的一端受到一个有初速度颗粒的撞击,导致颗粒连中产生稳定传播的应力波——高度非线性孤立波,该应力波的波长、波速以及幅值都能保持很好的稳定性,且遇到边界才会反射.孤立波是一种良好的信息载体,广泛应用于无损检测技术中.基于孤立波的特性,研究高度非线性孤立波与弹性大板耦合作用,基于赫兹定律和板的内在非弹性理论,推导出晶体链与大板的耦合微分方程组.用龙格库塔法求解该微分方程组,得到颗粒链中各颗粒的位移、速度曲线.通过分析回弹波出现的时间、回弹波所携带的能量以及模量、厚度、重力等对孤立波的影响,发现反射孤立波对大板的弹性模量和厚度尤为敏感,此外,颗粒链的摆放对整个耦合过程也有影响.研究的结果为孤立波对结构体的无损探伤提供了理论依据,该技术可实现对结构体的快速检查和可控性研究.     

圆柱形弹性壳液耦合系统大幅低频重力波的解析分析

以圆柱形弹性壳液耦合系统为研究对象,对已建立的多自由度非线性振动方程用平均法求其近似解析解,得到了方程组的一次近似定常解结果,由此可知当液固耦合系统受到激振力足够大,且激振频率处于弹性壳体的固有频率和重力波频率之和的一个窄频带范围时,液面以高频为主的振动会转化为以低频为主的大幅重力波运动,这是一种组合共振现象.     

Solitary Waves to Assess the Internal Pressure and the Rubber Degradation of Tennis Balls

Experimental Mechanics - Rubber is a material present in many commodities, including tennis balls. The characteristics of tennis balls are specified by the International Tennis Federation and are...     

Simulation of Traveling Solitary Plane Waves

The possibility of propagation of solitary plane waves with the Whittaker profile in materials with a microstructure (composites) is discussed. Solitary waves are defined as aperiodic smooth waves with an initial profile that is equal to zero everywhere except for some finite interval. Functions with indices 0.0, 0.1, –1/4, and 1/4 are chosen for computer simulation. It is observed that with some restrictions on the time or distance of propagation in the material, two modes of the traveling wave with the Whittaker profile and different phase-dependent phase velocities propagate simultaneously. The discussion section focuses attention on the conditions of blanking of the second mode for small values of the phase     

A Multiplicity Result for Solitary Gravity-Capillary Waves in Deep Water via Critical-Point Theory

. (Accepted May 29, 1998)     

Stability and Instability of Fourth-Order Solitary Waves

We study ground-state traveling wave solutions of a fourth-order wave equation. We find conditions on the speed of the waves which imply stability and instability of the solitary waves. The analysis depends on the variational characterization of the ground states rather than information about the linearized operator.     

Pressure Beneath a Solitary Water Wave: Mathematical Theory and Experiments

We study the pressure beneath a solitary water wave propagating in an irrotational flow at the free surface of water with a flat bed. The investigation is divided into two parts. The first part concerns a rigorous nonlinear study of the governing equations for water waves. We prove that the pressure in the fluid beneath a solitary wave strictly increases with depth and strictly decreases horizontally away from the vertical line beneath the crest. The second part of the paper describes an experimental study. Excellent agreement is found to exist between the theoretical predictions and the measurements. Our conclusions are also supported by numerical simulations.     

A Dimension-Breaking Phenomenon for Water Waves with Weak Surface Tension

It is well known that the water-wave problem with weak surface tension has small-amplitude line solitary-wave solutions which to leading order are described by the nonlinear Schrödinger equation. The present paper contains an existence theory for three-dimensional periodically modulated solitary-wave solutions which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction; they emanate from the line solitary waves in a dimension-breaking bifurcation. In addition, it is shown that the line solitary waves are linearly unstable to long-wavelength transverse perturbations. The key to these results is a formulation of the water wave problem as an evolutionary system in which the transverse horizontal variable plays the role of time, a careful study of the purely imaginary spectrum of the operator obtained by linearising the evolutionary system at a line solitary wave, and an application of an infinite-dimensional version of the classical Lyapunov centre theorem.     

Establishment of Waves Generated by a Pulsating Source in a Finite-Depth Fluid

A correct solution of Sretenskii’s plane problem of a source pulsating in a finite-depth fluid is derived. The solution is found using generalized functions as a limit in the infinite future of a wave regime generated by a source which starts to execute pulsations in a fluid initially at rest at a certain moment of time.     

Problem of Breakdown of Solitary Waves and Discontinuities

The evolution of initial data of the solitary-wave type with time is investigated numerically. The solitary wave amplitude decreases due to the generation of short-wave radiation. This solution is interpreted as the solution with a discontinuity qualitatively analogous to the solution of the problem of the breakdown of an arbitrary discontinuity in dissipationless systems. The solitary wave amplitude reduction rate is estimated, first for a generalized Korteweg-de Vries equation and then for plasma waves. Features of the investigation are analyzed for cold and hot-electron plasmas.     

水下散体颗粒滑坡产生海啸波的实验研究

海底滑坡海啸作为仅次于地震海啸的重要海啸形态,常常发生于靠近海岸带的陆架和陆坡等区域,且由于其近岸灾害影响显著、物理机制复杂、难以监测和预警等特点,给靠近海岸带的居民造成了巨大的生命财产损失．关于水上滑坡海啸的实验研究相对比较充分,而对水下滑坡海啸和部分淹没滑坡海啸的研究则相对较少．本研究在大型波浪水槽中进行了斜坡上散体水下滑坡激发海啸的机理实验,研究了滑坡体的滑移距离、海啸波的形成过程以及海啸波的幅值、波速、非线性等特征,系统地认识了滑坡体质量、水深以及滑坡角度对滑坡海啸波高的影响规律,并提出了改进的水下滑坡海啸特征参数经验公式．