拟Hamilton系统的随机平均

为拟Hamilton系统(小阻尼与弱随机激励作用下的Hamilton系统)提出了随机平均法,并就可积与不可积、共振与非共振情形进行了讨论.指出,现有的标准随机平均法与能量包线随机平均法为提出的拟Hamilton系统随机平均法的特殊情形.数例结果表明本方法是有效的.     

平方守恒系统与Hamilton系统

通过深入研究线性情形平方守恒系统与Hamilton系统在一定条件下的相互转换关系,揭示了平方守恒格式与辛格式之问内在和外在的联系,为辛格式在平方守恒系统以及平方守恒格式在Hamilton系统中的推广应用架起了桥梁,并打下了坚实的理论基础.     

Mather定理和平面Hamilton系统的Aubry-Mather集 *

推广了Mather关于正定Lagrange系统Aubry-Mather集在一维情形的存在性定理 ,同时给出了对平面Hamilton系统的一些应用.     

海洋表面波约化Hamilton方程的新发展:从小幅波到有限幅波的推广

海洋表面波的3-波至5-波约化Hamilton方程由于其对称多项式简化结构以及保能量等独特优点,得到广泛应用.但是,据相关近似假设,其适用范围局限于波陡很小的弱非线性波.于是进一步探讨下述推广问题: 对一定范围内的有限幅非线性波,在足够精确意义上是否也能获得具对称多项式简化结构的约化Hamilton方程？由于涉及复杂非线性强耦合,在该重要方面至今尚未取得进展.提出基于Chebyshev(切比雪夫)多项式逼近处理精确水波方程强非线性耦合的新简化途径,导出具对称多项式简化结构的新约化Hamilton方程.新结果将波数与波陡之积为小量的弱非线性情形拓广到该积直至1.035的非线性情形.分析表明,在该范围内新结果的误差不超过5%,特别,当前述积邻近于0.9时新结果给出精确结果.     

线性Hamilton系统边值问题的保辛数值方法

以Hamilton系统的正则变换和生成函数为基础研究线性时变Hamilton系统边值问题的保辛数值求解算法.根据第二类生成函数系数矩阵与状态传递矩阵的关系,构造了生成函数系数矩阵的区段合并递推算法,并进一步将递推算法推广到线性非齐次边值问题中;然后利用生成函数的性质将边值问题转化为初值问题,最后采用初值问题的保辛算法求解以达到整个Hamilton系统保辛的目的.数值算例表明该方法能够有效地求解线性齐次与非齐次问题,并能很好地保持Hamilton系统的固有特性.     

渐近线性二阶Hamilton系统解的存在性和多重性

在实际问题中存在着Neumann边值情形.为实际需要,运用指标理论和Morse理论研究了渐近线性二阶Hamilton系统在这种情形下解的存在性和多重性问题。     

一类孤立子系统的Hamilton结构及Liouville可积性

本文给出了一个2×2谱问题及其相应的孤子族,并利用此孤子族的Lenard算子对的性质,证明了该系统是具有Bi-Hamilton结构和Multi-Hamilton结构的广义Hamilton系统,进一步给出其Liouville可积性的证明.此外,值得提出的是此系统可约化为广义TD族、TD族和广义C-KdV族、C-KdV族等,并得到了该孤子族的Hamilton泛函与守恒密度之问的一一对应关系.     

变分与无限维系统的高精度辛格式

1.引 言 冯康和他的研究小组提出的生成函数法[1]系统地解决了象二体问题这样地有限维Hamil-ton系统辛算法的构造问题,该方法也可以自然地推广到无限维Hamilton系统[2].首先在空间方向进行离散,例如采用差分或谱离散,得到有限维Hamilton系统,然后再采用生成函数法离散该系统.这样得到的辛格式是整个一层的格式,对于研究格式的局部性质如多辛性质[3],局部能量守恒性质[5]就相当困难.     

Hamilton等参元与智能叠层板的半解析法

根据压电材料修正后的Hellinger-Reissner(H-R)变分原理,建立了各向异性压电材料4节点Hamilton等参元的一般形式．为智能叠层板自由振动问题和带有压电块的叠层悬臂梁的瞬态响应等问题提出了一种新的半解析法．数学模型的基本步骤:将压电层和主体层看成独立的三维体,在平面内离散各层,分别建立各层的方程;根据主体层和压电层在连接界面上广义应力和广义位移的连续条件,联立主体层和压电层的方程得到全结构的控制方程．等参元不限制智能板侧面的几何边界形状、板的厚度和层数,有广泛的应用领域．     

缓变深度分层流体中的准周期波和准孤立波

本文讨论具缓变深度二流体系统中的非线性波,该系统由一不规则底部与一水平固壁间的两层常密度无粘流体所组成.文中用约化摄动法导出了所考虑模型的变系数Korteweg-de Vries方程,并用多重尺度法求出了该方程的近似解,发现底部固壁的不规则变化将产生所谓准周期波和准孤立波.它们的周期、波速和波形将发生缓慢变化,文中给出了准周期波的周期随深度的变化关系式以及准孤立波波幅、波速随深度的变化关系式,底部水平情形和单层流体情形可看成是本文的特例.     

Hamiltonian model for coupled surface and internal waves in the presence of currents

We examine a two dimensional fluid system consisting of a lower medium bounded underneath by a flatbed and an upper medium with a free surface. The two media are separated by a free common interface. The gravity driven surface and internal water waves (at the common interface between the media) in the presence of a depth-dependent current are studied under certain physical assumptions. Both media are considered incompressible and with prescribed vorticities. Using the Hamiltonian approach the Hamiltonian of the system is constructed in terms of ‘wave’ variables and the equations of motion are calculated. The resultant equations of motion are then analysed to show that wave–current interaction is influenced only by the current profile in the ‘strips’ adjacent to the surface and the interface. Small amplitude and long-wave approximations are also presented.     

Hamiltonian spatial structure for three-dimensional water waves in a moving frame of reference

Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is, motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian. The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws) is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for capillary-gravity waves, showing how new waves can be found in this framework.     

Hamiltonian Boussinesq formulation of wave–ship interactions

In this paper a new approach is described for the fully nonlinear treatment of the dynamic wave–ship interaction for potential flows. A major reduction of computational complexity is obtained by describing the fluid motion in horizontal variables only, the surface elevation and the potential at the surface. In such Boussinesq type of equations, the internal fluid motion is not calculated, but modeled in a consistent approximative way. The equations for the wave–ship interaction are based on a Lagrangian variational principle, leading to the formulation of the coupled system as a Hamiltonian system. With the ship position and orientation as canonical coordinates, the canonically conjugate momentum variables are the sum of the ship momemta and the fluid momenta. A beneficial consequence of this is that the momentum exchange between fluid and ship will be described without the need to calculate the pressure, which simplifies the numerical implementation of the equations considerably. Provided that the potentials with mixed Dirichlet–Neumann data can be calculated, the presented ship dynamics can be inserted in existing free surface flow solvers.     

Hamiltonian long‐wave expansions for free surfaces and interfaces

The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature. From the formulation, we develop a Hamiltonian perturbation theory for the long‐wave limits, and we carry out a systematic analysis of the principal long‐wave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker (28), Benjamin (3, 4), Ono (26), and many others. Our considerations include the Boussinesq and Korteweg–de Vries (KdV) regimes over finite‐depth fluids, the Benjamin‐Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long‐wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface. Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa (6, 7), we also derive novel systems of nonlinear dispersive long‐wave equations in the large‐amplitude, small‐slope regime. Our formulation of the dynamical free‐surface, free‐interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations. © 2005 Wiley Periodicals, Inc.     

二层流体中沿任意路径运动的奇点解析解*

本文求解了二层流体存在时在上层或下层流体中沿任意路径运动的奇点的速度势.流体的深度为有限或者无限.计入上层流体的影响后,流体交界面不再是等压力面或零压力面.文中给出了这种情况下的一系列奇点基本解的解析表达式.     

A Quasi‐Planar Model for Gravity‐Capillary Interfacial Waves in Deep Water

A dynamical model equation for interfacial gravity‐capillary (GC) waves between two semi‐infinite fluid layers, with a lighter fluid lying above a heavier one, is derived. The model proposed is based on the fourth‐order truncation of the kinetic energy in the Hamiltonian of the full problem, and on weak transverse variations, in the spirit of the Kadomtsev‐Petviashvilli equation. It is well known that for the interfacial GC waves in deep water, there is a critical density ratio where the associated cubic nonlinear Schrödinger equations changes type. Our numerical results reveal that, when the density ratio is below the critical value, the bifurcation diagram of plane solitary waves behaves in a way similar to that of the free‐surface GC waves on deep water. However, the bifurcation mechanism in the vicinity of the minimum of the phase speed is essentially similar to that of free‐surface gravity‐flexural waves on deep water, when the density ratio is in the supercritical regime. Different types of lump solitary waves, which are fully localized in both transverse and longitudinal directions, are also computed using our model equation. Some dynamical experiments are carried out via a marching‐in‐time algorithm.     

Wave structure interaction problems in three-layer fluid

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.     

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 0²‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 0²‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalized Darcy–Oseen resolvent problem

In this paper, we study the well‐posedness of a coupled Darcy–Oseen resolvent problem, describing the fluid flow between free‐fluid domains and porous media separated by a semipermeable membrane. The influence of osmotic effects, induced by the presence of a semipermeable membrane, on the flow velocity is reflected in the transmission conditions on the surface between the free‐fluid domain and the porous medium. To prove the existence of a weak solution of the generalized Darcy–Oseen resolvent system, we consider two auxiliary problems: a mixed Navier–Dirichlet problem for the generalized Oseen resolvent system and Robin problem for an elliptic equation related to the general Darcy equations. © 2016 The Authors Mathematical Methods in the Applied Sciences Published by John Wiley & Sons Ltd.