SH波对一维六方准晶中快速传播裂纹的散射

利用积分变换技术,结合Copson方法,研究了SH波对一维六方准晶中快速传播裂纹的散射问题.通过对偶积分方程求解,得到声子场和相位子场应力、位移的解析表达式,同时给出了裂纹尖端标准动应力强度因子的定义.数值算例表明随着裂纹传播速度、裂纹长度、入射角、入射波频率增大,裂纹尖端标准动应力强度因子也在增大.当裂纹长度的增大到一定值时,裂纹的传播速度对标准动应力强度因子影响逐渐减小.研究结果在工程材料应用中有一定的参考价值.     

Poisson系统的辛结构

当Poisson系统中的Poisson矩阵是非常数时,经典的辛方法如辛Runge-Kutta方法,生成函数法一般不能保持Poisson系统的Poisson结构,利用非线性变换可把非常数Poisson结构转化成辛结构,然后任意阶的辛方法可以长时间计算Poisson系统的辛结构．自由刚体问题中Euler方程被转换成辛结构并用辛中点格式进行数值求解,数值结果给出了这种非线性变换的有效性．     

非线性弹性杆波动方程的显式精确解北大核心CSCD

应用sine-cosine方法对非线性弹性杆波动方程进行了求解,得到了该方程的一些新的周期波解和孤波解(材料常数n为不等于1的常数).对部分结果通过数学软件得到了解的图像,获得的结果有助于非线性弹性杆中孤波存在性问题的进一步研究.     

板列弯曲振动及功率流分析的辛空间波传播方法

基于波传播理论,在辛空间下研究了由矩形薄板组成的板列结构的自由波属性以及受迫振动问题.通过将薄板弯曲振动控制方程导入辛对偶体系,得到了薄板波传播参数以及各阶波形的辛解析解.根据波在各板之间的传播、反射以及透射关系和叠加原理得到问题的解.给出了辛空间-波传播框架下各板动能、应变能以及板间功率流的计算表达式.相比传统波传播方法,该方法具有不受边界条件限制以及能够给出波模态辛解析解的特点.以一个三板组合结构为算例,通过与ABAQUS程序得到的有限元参考解进行对比,验证了所提出方法的高效性与精确性.由于完全基于理性推导,不涉及任何试函数的引入,因此该方法也可推广应用于由其他类型板(如中厚板、层合板等)组合的板列结构动力响应分析问题.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

饱和地基上弹性基础的竖向振动特性研究

采用解析的方法研究了饱和地基上受一简谐竖向荷载作用下弹性基础的动力响应．在分析中,首先利用积分变换技术获得了饱和介质基本控制方程的变换解,然后基于基础-半空间完全放松接触、半空间表面完全透水或不透水的假设,建立了该动力混合边值问题的对偶积分方程,并把该对偶积分方程进一步化为易于数值求解的第二类Fredholm积分方程A·D2文末数值算例给出了动力柔度系数、位移和孔隙水压力随振动频域和土-基础体系物理力学参数特性的变化曲线．结果表明:饱和地基上弹性基础的动力响应完全不同于饱和地基上刚性圆板的动力响应．所用方法可用于研究波的传播、土-结构动力相互作用等许多问题．     

非线性梁结构的参数振动稳定性及其主动控制

采用压电材料研究了参数激励非线性梁结构的运动稳定性及其主动控制,通过速度反馈控制算法获得主动阻尼,利用Hamilton原理建立含阻尼的立方非线性运动方程,采用多尺度方法求解运动方程获得稳定性区域．通过数值算例,分析了控制增益、外激振力幅值等因素对稳定性区域和幅频曲线特性的影响．分析表明:控制增益增大,结构所能承受的轴向力也增大,在一定范围内结构的主动阻尼比也增加;随着控制增益的增大,响应幅值逐渐降低,但所需的控制电压存在峰值点．     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.     

随机地震激励下水中悬浮隧道的动力响应

将张力腿定位的水中悬浮隧道结构简化为弹性简支梁和弹性支撑刚性梁的叠加,基于Euler(欧拉)梁理论给出悬浮隧道管段受迫振动时的运动方程,等效线性化处理动力方程非线性项;采用虚拟激励模拟随机地震输入,数值模拟平稳随机地震下水中悬浮隧道管体的动力响应,给出管段的位移功率谱.通过位移功率谱分析表明:随着消能连接装置阻尼系数和张力腿弹簧系数的增大,管段的动力响应减弱,其中张力腿的刚度对悬浮隧道管体的震动影响较为显著.     

Sine-Gordon方程的多辛Leap-frog格式

非线性发展方程由于具有多种形式的解析解而吸引着众多的研究者,借助多辛保结构理论研究了Sine-Gordon方程的多辛算法．利用Hamilton变分原理,构造出了sine-Gordon方程的多辛格式;采用显辛离散方法得到了Leap-frog多辛离散格式,该格式满足多辛守恒律;数值结果表明leap-frog多辛离散格式能够精确地模拟sine-Gordon方程的孤子解和周期解,模拟结果证实了该离散格式具有良好的数值稳定性．     

Bifurcations and exact traveling wave solutions of the equivalent complex short-pulse equations

In this paper, we study the traveling wave solutions for a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrodinger (NLS) equation in the ultrashort-pulse regime. The corresponding traveling wave systems of the equivalent complex short-pulse equations are two singular planar dynamical systems with four singular straight lines. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.     

Wave propagation properties in oscillatory chains with cubic nonlinearities via nonlinear map approach

Free wave propagation properties in one-dimensional chains of nonlinear oscillators are investigated by means of nonlinear maps. In this realm, the governing difference equations are regarded as symplectic nonlinear transformations relating the amplitudes in adjacent chain sites (n, n + 1) thereby considering a dynamical system where the location index n plays the role of the discrete time. Thus, wave propagation becomes synonymous of stability: finding regions of propagating wave solutions is equivalent to finding regions of linearly stable map solutions. Mechanical models of chains of linearly coupled nonlinear oscillators are investigated. Pass- and stop-band regions of the mono-coupled periodic system are analytically determined for period-q orbits as they are governed by the eigenvalues of the linearized 2D map arising from linear stability analysis of periodic orbits. Then, equivalent chains of nonlinear oscillators in complex domain are tackled. Also in this case, where a 4D real map governs the wave transmission, the nonlinear pass- and stop-bands for periodic orbits are analytically determined by extending the 2D map analysis. The analytical findings concerning the propagation properties are then compared with numerical results obtained through nonlinear map iteration.     

多尺度辛格式求解复杂介质波传问题

在哈密顿体系中引入小波分析,利用辛格式和紧支正交小波对波动方程的时、空间变量进行联合离散近似,构造了多尺度辛格式——MSS（Multiresolution Symplectic Scheme）．将地震波传播问题放在小波域哈密顿体系下的多尺度辛几何空间中进行分析,利用小波基与辛格式的特性,有效改善了计算效率,可解决波动力学长时模拟追踪的稳定性与逼真性．     

Symplectic and multi-symplectic wavelet collocation methods for two-dimensional Schrödinger equations

In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods.     

M.Wadati可积非线性发展方程的精确行波解

用平面动力系统方法研究由M.Wadati提出的一类可积非线性发展方程的精确行波解,获得了该方程的扭波、反扭波解,周期波解和不可数无穷多光滑孤立波解的精确的参数表达式,以及上述解存在的参数条件．     

Singular periodic waves of an integrable equation from short capillary-gravity waves

The effects of parabola singular curves in the integrable nonlinear wave equation are studied by using the bifurcation theory of dynamical system. We find new singular periodic waves for a nonlinear wave equation from short capillary-gravity waves. The new periodic waves possess weaker singularity than the periodic peakon. It is shown that the second derivatives of the new singular periodic wave solutions do not exist in countable number of points but the first derivatives exist. We show that there exist close connection between the new singular periodic waves and parabola singular curve in phase plane of traveling wave system for the first time.     

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.     

Peakon, pseudo-peakon, loop, and periodic cusp wave solutions of a three-dimensional 3DKP(2, 2) equation with nonlinear dispersion

In this paper, we study the three-dimensional Kadomtsev-Petviashvili equation (3DKP(m, n)) with nonlinear dispersion for m=n=2. By using the bifurcation theory of dynamical systems, we study the dynamical behavior and obtain peakon, pseudo-peakon, loop and periodic cusp wave solutions of the three-dimensional 3DKP(2, 2) equation. The parameter expressions of peakon, pseudo-peakon, loop and periodic cusp wave solutions are obtained and numerical graph are provided for those peakon, pseudo-peakon, loop and periodic cusp wave solutions.