约化摄动法和非线性波远场分析

一、引言如所周知,摄动法(特别是奇异摄动法)在研究弱非线性波方面有着广泛的应用[1—5]。其中,近年来发展形成的约化摄动法已经成了分析各种非线性波远场的有力工具[6—8]。约化摄动法的实质是,对于一般的描述非线性波的复杂方程组,通过适当的坐标变形和摄动展开,在一阶近似下,把方程组约化成较为简单可解的单个非线性方程(例如Burgers方程、Korteweg-de Vries方程、非线性Schrodinger方程等),从而可以分析远离波的相互作用区的远场。      

Stationary-state solutions to the rotation of solid bodies with liquid-filled cavities and their stability

In this paper,we discuss all the possible equilibrium states of axi-symmetrical-solidbodies with liquid-filled cavities rotating around.fixed axes according to the extremumconditions on the potential energy, and conclude that there exists a unique stable final-statesolution, for which the system uniformly rotates around its vertical symmetrical axis,forboth the inverted and suspended ones. And then applying the Lyapunoy direct approach fora continuous system.we investigate the stability of the rotating systems subject to largedisturbances. In addition, we describe an interesting analogue between the rotation of asolid body with a liquid-filled cavity in the inverted case and the motion of a small ball in aspinning spherical bowl. The results obtained herein theoretically provide an evidence of thereality of the secular stability.     

含稀疏分布杂质的复合材料中的均匀化稳恒热传导方程

本文应用双空间尺度法导出了含稀疏分布椭圆柱形杂质的复合材料柱中的均匀化稳恒热传导方程,求得了等效导热系数的具体形式,并指出,当杂质枉截面单向分布时,宏观热传导是各向异性的,而当杂质枉截面按方向均匀分布时,宏观热传导是各向同性的.     

Quasi-periodic waves and quasi-solitary waves in stratified fluid of slowly varying depth

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper. The model considered here consists of a two-layer incompressible constantdensity inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall. The Korteweg-de Vries (KdV) equation with varying coefficients is derived with the aid of the reductive perturbation method. By using the method of multiple scales, the approximate solutions of this equation are obtained. It is found that the unevenness of bottom may lead to the generation of so-called quasi-periodic waves and quasisolitary waves, whose periods, propagation velocities and wave profiles vary slowly. The relations of the period of quasiperiodic waves and of the amplitude, propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented. The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.Project Supported by National Natural Science Foundation of China.     

QUASI-PERIODIC WAVES AND QUASI-SOLITARY WAVES IN STRATIFIED FLUID OF SLOWLY VARYING DEPTH

The nonlinear waves in a stratified fluid of slowly varying depth are investigated in this paper.The model considered here consists of a two-layer incompressible constant-density inviscid fluid confined by a slightly uneven bottom and a horizontal rigid wall.The Korteweg-de Vries(KdV)equation with varying coefficients is derived with the aid of the reductive perturbation method.By using the method of multiple scales,the approximate solutions of this equation are obtained.It is found that the unevenness of bottom may lead to the generation of socalled quasi-periodic waves and quasi-solitary waves,whose periods,propagation velocities and wave profiles vary slowly.The relations of the period of quasi-periodic waves and of the amplitude,propagation velocity of quasi-solitary waves varying with the depth of fluid are also presented.The models with two horizontal rigid walls or single-layer fluid can be regarded as particular cases of those in this paper.     

SOLITARY WAVES AT THE INTERFACE OF A TWO-LAYER FLUID

In this paper,we discuss the solitary waves at the interface of a two-layer incompressible inviscid fluid confined by two horizontal rigidwalls,taking the effect of surface tension into account.First of all,we establish the basic equations suitable for the model considered,and hence derive the Korteweg-de Vries(KdV)equation satisfied bythe first-order elevation of the interface with the aid of the reduc-tive perturbation method under the approximation of weak dispersion.Itis found that the KdV solitary waves may be convex upward or downward.It depends on whether the signs of the coefficients α and μof the KdVequation are the same or not.Then we examine in detail two criticalcases,in which the nonlinear effect and the dispersion effect cannotbalance under the original approximation.Applying other appropriateapproximations,we obtain the modified KdV equation for the criticalcase of first kind(α=0),and conclude that solitary waves cannot existin the case considered as μ>0,but may still occur as μ<0,be     

TWO-DIMENSIONAL ALGEBRAIC SOLITARY WAVE AND ITS VERTICAL STRUCTURE IN STRATIFIED FLUID

IntroductionItis known thatthere exists a special kind of solitary waves—the algebraic solitary waveas the disturbance range is close to the scale of vertical variation of density in thecontinuously stratified fluid.The algebraic solitary wave is governe…     

Characteristic Parameters of a Wide Cluster in a Higher-Order Traffic Flow Model

Nonlinear weak solution theory is applied to determine the parameters of a wide cluster in an ＂anisotropic＂ higher-order traffic flow model. These parameters are the maximal and minimal densities and the travelling wave speed in the solution structure, Numerical experiments show that the convergent simulation results arc in good agreement with those obtaincd from the analytical expressions.     

谈镐生简介

谈镐生教授是我国著名力学家,现任中国科学院学部委员、学位委员,中国科学院力学研究所副所长兼学术委员会主任,中国科技大学力学系主任、研究生院物理数学部副主任,中国力学学会常务理事.由于他早年在空气动力学和湍流理论方面的成就,近年来又...     

用数学武装工程科学

人们常说,研究数学的主要目的之一是为物理学家和工程师们提供解决实际问题的工具。从数学的发展史看来,事实很清楚,许多重大的数学发现是在了解自然规律的迫切要求下应运而生的,许多数学方法是由主要对实际应用感兴趣的人创立的。然而,每个真正的数学家都会感到,把数学研究局限于考察那些有直接应用的问题,对这位“科学的皇后”来说未免有点不公道,事实上,这位“皇后”的虔诚的歌颂者对于把他们的女主人贬黜为她的比较注重实际的、一时较为显赫的姐妹的“侍女”,经常感到忿忿不平。