The uniqueness and existence of solution of the characteristic problem on the generalized KdV equation

The generalized KdV equationu ₁+auu_a+μu_a3+eu_a5=0^[1] is a typical integrable equation. It is derived studying the dissemination of magnet sound wave in cold plasma^[2], the isolated wave in transmission line^[3], and the isolated wave in the boundary surface of the divided layer fluid^[4]. For the characteristic problem of the generalized KdV equation, this paper, based on the Riemann function, designs a suitable structure, then changes the characteristic problem to an equivalent integral and differential equation whose corresponding fixed point, the above integral differential equation has a unique regular solution, so the characteristic problem of the generalized KdV equation has a unique solution. The iteration solution derived from the integral differential equation sequence is uniformly convegent in .     

AGAIN DISCUSSING ABOUT SINGULAR PERTURBATION OF GENERAL BOUNDARY VALUE PROBLEM FOR HIGHER ORDER ELLIPTIC EQUATION CONTAINING TWO-PARAMETER

In this paper using the method of "The Two-Variable Expansion Procedure" we again discuss the construction of asymptotic expression of solution of general boundary value problem for higher order ellitptic equation containing two-parameter whose boundary condition is more general than [1]. We give asymptotic expression of solution as well as the estimation corresponding to the remainder term.     

KdV-Burgers方程的级数解

给出ＫｄＶ－Ｂｕｒｇｅｒｓ方程的有界行波解的精确级数解,采用Ａｄｏｍｉａｎ算子分解法分别求行二个区域ζ〈０和ζ〉０的级数解,然后利用对接连续条件构成整体级数解。所得级数解能精确满足对接连续条件,并由此得到确定级数的系数递推公式,无需解非红性高阶代数方程组。与某些精确解及其它方法比较,计算简捷具在对接点处是收剑的。对某些非线性波动现象的研究,可作为计算和分析的数学依据。     

Axially symmetric long waves on the surface of a varying-depth basin

The axially symmetric Korteweg-de Vries (KdV) equation for the case of a constant-depth basin was obtained in [1]. In the present paper we derive the axially symmetric KdV equation for a varying-depth basin. Conditions are shown for the equation obtained, under which the asymptotic behavior of its solution is described by an equation of the form $$u_t + uu_x + u_{xxx} = 0,$$ whose asymptotic behavior is well known [2].     

Maximum amplitude of modulated wavetrain

For water wave predictions, it is of practical importance to know the size of the biggest wave that emerges from a given uniform wavetrain when it experiences Benjamin-Feir's modulational instability. As far as we are aware, the series of experiments performed by Su and Green [5] are the only work that has pursued this problem so far. In this work their experimental result is compared with theoretical predictions given by the nonlinear Schrödinger equation as well as with the numerical simulation of the fully nonlinear water wave equation. The result of the fully nonlinear simulation shows that the height of the biggest wave reaches more than 3.5 times the initial amplitude of the wavetrain at the maximum modulation when the initial steepness a₀k₀ of the wavetrain is 0.125.     

SH0导波在粘接结构中传播时的相位变化1）

对粘接结构进行超声导波无损检测与评估是一个有挑战性的前沿性课题.针对此问题,研究了SH₀导波在界面为理想连接的三层板状粘接结构中传播时的相位变化情况.首先基于波传播的控制方程,建立了粘接结构中反射和透射SH₀导波相对于入射SH₀导波的相位差解析模型.然后利用数值模拟计算了铝/环氧树脂/铝粘接结构中反射和透射SH₀导波的相位差曲线.最后分析了入射角度和频厚积的变化对反射和透射SH₀导波相位差的影响.结果表明,对于具体的粘接结构,反射和透射SH₀导波在其中传播时的相位差变化主要依赖于入射角度、频率等参数.在特定的频厚积下,当声波水平入射时,反射和入射SH₀导波同相位.无论入射角度为多大,随着频厚积的增大,反射SH₀导波的相位差曲线均会产生周期性谐振.对于透射SH₀导波,当声波垂直入射时,其相位差曲线的改变无规律可循;但是随着入射角度逐渐增大,透射SH₀导波的相位差曲线逐渐变规则.所得结果可为实验时研究板状粘接结构中SH₀导波的传播特性以及提取SH₀导波回波中的有用信息和定位提供一定的理论指导.     

集中力作用下的两相饱和介质位移场Green函数

以复模两相饱和介质Biot动力学方程为基础,根据该方程D'Alembert解的Fourier变换所属的Homholtz方程特性,由Biot方程解的相容性条件及δ函数性质较好地解决了快、慢纵波位势的耦合问题.较为简便地得到了两相饱和介质在集中力作用下低频(ω＜ωc)时的频域和时域的Green函数.     

Steady wave regimes in a film of viscous liquid flowing down a vertical wall

A study is made of the steady wave solutions of the nonlinear third-order differential equation [1] that describes the behavior of the wave boundary of a thin film of viscous liquid flowing down a vertical wall. It is shown that for long waves of small amplitude the general equation can be reduced [2] to a form containing a unique dimensionless parameter. A qualitative investigation is made of the behavior of the integral curves and the types of the singular points in the phase space. It is shown that a solitary wave exists for discrete values of the dimensionless parameter. A numerical solution is obtained. The structure of the jump in the thickness of the film is investigated qualitatively. Numerical solutions of nonmonotonic structure are obtained for different parameters.     

Solitary wave solutions to higher-order traffic flow model with large diffusion

This paper uses the Taylor expansion to seek an approximate Korteweg- de Vries equation （KdV） solution to a higher-order traffic flow model with sufficiently large diffusion. It demonstrates the validity of the approximate KdV solution considering all the related parameters to ensure the physical boundedness and the stability of the solution. Moreover, when the viscosity coefficient depends on the density and velocity of the flow, the wave speed of the KdV solution is naturally related to either the first or the second characteristic field. The finite element method is extended to solve the model and examine the stability and accuracy of the approximate KdV solution.     

Experimental investigation of cylindrically diverging solitons in an electric lattice

The results of an experimental investigation of cylindrical solitons in a two-dimensional electric LC-lattice are given. It is shown that in the continuum limit, propagation of cylindrical waves far from the center of symmetry in such a lattice may be described for each ray tube by a known modification of the Korteweg-de Vries equation which takes account of the cylindrical divergence. The dispersion term in this equation depends on the direction of wave propagation relative to the direction of the main axes of the lattice. Formation of solitons from non-soliton-shaped pulses was observed. The variations of soliton amplitude and duration with distance have been determined. They agree well with the numerical calculations by Maxon & Viecelli [2] and Dorfman [9]. Comparison of the obtained experimental data with the known theoretical laws of amplitude attenuation for diverging solitons [2, 12, 14] seems to favor the validity of the law A r^−2/3 rather than A r^−1/2.     

分层流体中内孤立波在台阶上的反射和透射

基于匹配渐近展开和格林函数的方法,研究了两层流体系统中内孤立波在台阶地形上透射、 反射及其分裂的演化特征. 通过保角变换和求解奇异Fredholm积分方程,获得了反映地形 效应对Boussinesq方程影响的约化边界条件,藉此建立了KdV演化方程的``初值'问题, 根据散射反演理论获得了反射波和透射波的解析表达式. 分析结果表明：上下流体层的厚度 比、密度比以及台阶高度对于反射和透射波振幅及其分裂具有显著的影响. 尤其当上层流体 厚度小于下层厚度时,由于存在临界点,在其附近反射波的幅值随台阶高度的演化由单调增 变为单调减,透射波的幅值由单调减变为单调增;上台阶的反射波与入射波反相,其最大幅 值可达到入射波的数倍;此外,下台阶反射波也可发展为单支孤立波,它区别于单层流体中 反射波仅为衰减的振荡波列.     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

论Hertz接触问题解的不唯一性

本文根据积分方程的理论,对Hertz接触问题的积分方程的特点进行了分析,指出该问题的解不是唯一的。     

横观各向同性三维热弹性力学通解及其势理论法

通过引入两个位移函数，对用位移表达的运动平衡方程作了简化．利用算子理论，严格地导出了横观各向同性非耦合热弹性动力学问题的通解．对于静力学问题，通解的形式可进一步简化成用4个准调和函数来表示．具体考察了横观各向同性体内平面裂纹上下表面有对称分布温度作用的问题，推广了势理论方法，导出了一个积分方程和一个微分-积分方程．针对币状裂纹表面受均布温度作用情形，给出了具体的解。     

孔板扰动对爆轰波胞格结构特性影响的实验研究

实验采用稳定预混气2H₂+O₂+3Ar及不稳定预混气C₂H₂+5N₂O和CH₄+2O₂,在圆形爆轰管内通过烟膜手段记录了爆轰波的胞格结构,得到了胞格尺寸与初始压力之间的关系式;研究了胞格结构在扰动上下游的变化过程,分析了胞格不稳定性对胞格结构特征的影响,获得了爆轰波经过扰动后重新恢复至平衡状态的特征尺度。结果表明:爆轰波经过扰动后,对于稳定预混气,在扰动下游主胞格结构变得不规则,没有出现次生胞格;对于不稳定预混气,扰动下游伊始爆轰波的次生模态被抑制,由于爆轰波自身的不稳定性,随后出现了局部爆炸点及精细胞格结构;爆轰波在扰动下游传播了一段距离后恢复至平衡状态,该长度在8～15倍之间的胞格尺寸范围内变化,并且随初始压力的变化趋势并不明显。研究结果反映出爆轰波经过孔板扰动后恢复至平衡态所需的长度与爆轰波流体动力学厚度相当。     

胆碱杂环二酸离子液体水润滑添加剂的制备及结构-性能关系研究

以胆碱和杂环二酸为原料,在去离子水中原位制备了添加剂[Ch]₂[Hdc],并研究了它们的摩擦学性能、腐蚀性、水生生物毒性与分子结构之间的构效关系. 研究发现,水溶液的运动黏度随着水中生成的[Ch]₂[Hdc]浓度的增加而增大,并且当[Ch]₂[Hdc]的分子结构对称性较低、极性较大时,水的黏度增加值相对较大. 这是由于添加剂分子极性较大时,分子间相互作用力较大,导致溶液的黏度增幅更大. 摩擦学性能测试发现,当[Ch]₂[Hdc]的浓度相对较低时,水溶液的减摩抗磨性能与[Ch]₂[Hdc]分子在摩擦副表面的吸附能力有关,分子极性较大,吸附能力较强的[Ch]₂[Hdc]可在摩擦副表面形成更为牢固的润滑保护膜,因而能有效改善水的减摩抗磨性能,反之则不能. 当[Ch]₂[Hdc]的浓度相对较高时,水溶液的减摩性能与其黏度呈反相关关系,抗磨性能则与其黏度呈正相关关系. 这是由于润滑剂黏度相对较大时,其内摩擦力较大,因而表现出较高的摩擦系数,即较差的减摩性能;然而,黏度相对较大的润滑剂则可以在摩擦副表面形成更为牢固的润滑保护膜,因而表现出较低的磨损体积,即较好的抗磨性能. 腐蚀试验结果表明,[Ch]₂[Hdc]可显著降低水对金属基底材料的腐蚀性. 毒性试验显示[Ch]₂[Hdc]对绿藻和海虾毒性远远小于传统离子液体L-B104.      

Geometrical nonlinear waves in finite deformation elastic rods

IntroductionSomenewphenomenaofnonlinearwavesinthesolidmediumsuchasshockwave ,solitarywaveetc.arepaidmoreattentiontoincreasinglybyresearchersbecausetheytakeonalotofimportantproperties.ItistheoreticallyanalyzedinRefs.[1 -6]thattheformationmechanismsofshockwaveandsolitarywaveintheelasticthinrodsaswellastheirpropagationproperties.TheexistenceofsolitarywaveintheelasticmediumsuchasarodandaplatehasbeenverifiedinRef.[7]byexperiments.Shockwaveandsolitarywavearesteadilypropagatingtraveling_wavesgenerat…     

Exact solutions to the Korteweg-de Vries-Burgers equation

We introduce a transormation which reduces the Korteweg-de Vries-Burgers (KdVB) equation, u_t + 2auu_x + 5bu_xx + cu_xxx = 0, to a quadratic form involving a new dependent variable and its partial derivatives. Exact solutions of the KdVB equation can be obtained by solving this equation. The exact form of the travelling wave solution to the KdVB equation is obtained in this paper, and its nature depends on the direction of propagation of the wave.