计及结构阻尼的等截面梁纯弯曲振动的非线性分析

针对考虑挠度微分方程中高阶项所引起的几何非线性及材料阻尼所引起的阻尼非线性的梁，建立了梁纯弯曲振动时的非线性运动方程。利用非线性理论对该非线性问题进行了研究，得到了周期解稳定和不稳定区域的分界线方程和频率响应方程，得到忽略梁非线性因素的条件，得到了梁挠度微分方程中高阶项所引起的几何非线性项具有软特性效应等四点结论。     

光纤中两个高阶变系数薛定谔方程的精确解

本文创造性地运用辅助方程方法研究了高阶变系数非线性偏微分方程的求解,其实质是基于常微分方程的解构造非线性偏微分方程的精确解。文章借助几个辅助常微分方程构造了两个高阶变系数非线性薛定谔方程的多个新型精确解,包括亮孤子、暗孤子以及单周期波解等,并推广了其中一个方程,给出了该方程的一些新型精确解。     

一类具偏差变元的高阶Liénard型方程的周期解

目前关于具偏差变元高阶Liénard型方程周期解的存在性的研究工作还很少,而且对其周期解的先验估计比较粗糙,因此本文基于更精确的先验估计,利用重合度理论中的连续定理,获得了一类具偏差变元的高阶Liénard型方程周期解存在性的若干新结论,所得结果推广了已有文献的相关结论,并使条件有所减弱。     

一类具偏差变元的高阶Liénard型方程的周期解

目前关于具偏差变元高阶Liénard型方程周期解的存在性的研究工作还很少，而且对其周期解的先验估计比较粗糙，因此本文基于更精确的先验估计，利用重合度理论中的连续定理，获得了一类具偏差变元的高阶Liénard型方程周期解存在性的若干新结论，所得结果推广了已有文献的相关结论，并使条件有所减弱。     

Davey-Stewartson Ⅰ方程新的精确周期解

Davey-Stewartson方程描述了有限深度的水中水波的运动,它的第一种类型称为(Davey-Stewartson I)是椭圆一双曲型方程。在物理学中,微分方程的精确解对考察非线性现象起着非常重要的作用,为了揭示Davey-Stewartson I方程的运动性质,本文研究它的精确周期解。应用F-代数方法并通过一个高阶辅助微分方程,获得了Davey-Stewartson I方程的一系列新的精确周期解,包括三角函数周期解,Jacobi椭圆函数周期解。     

具任意次非线性项的非线性Klein-Gordon方程孤波解的轨道稳定性

具任意次非线性项的非线性Klein-Gordon方程是一类非常重要的物理模型,它的孤波解的轨道稳定性有着很好的物理意义.本文利用抽象的Grillakis轨道稳定性理论和谱分析,讨论具任意次非线性项的非线性Klein-Gordon方程的孤波解的轨道稳定性.当非线性项的系数以及波速满足一定的条件时,得出了其钟状孤波解总是不...     

Davey-Stewartson I方程新的精确周期解

Davey-Stewartson方程描述了有限深度的水中水波的运动,它的第一种类型称为(Davey-Stewartson I)是椭圆一双曲型方程.在物理学中,微分方程的精确解对考察非线性现象起着非常重要的作用,为了揭示Davey-Stewartson I方程的运动性质,本文研究它的精确周期解.应用F-代数方法并通过一个高阶辅助微分方程,获得了Davey-Stewartson I方程的一系列新的精确周期解,包括三角函数周期解,Jacobi椭圆函数周期解.     

一类非线性jerk方程的改进两变量展开法

应用改进的两变量展开法求解非线性含有三次非线性项的三阶微分方程的近似频率和近似解析周期解。该方法结合了Lindstedt-Poincare方法与两变量展开法不仅可以适用于弱非线性振动问题的求解而且还可以适用于强非线性振动问题的求解。文中以一个不含速度线性项的非线性jerk方程作为例子分析并得到二阶近似周期和二阶近似解析周期解,与数值方法给出的“精确”周期解比较,二阶近似解析周期解比一阶近似解析周期解要精确得多。结果表明,改进的两变量展开法能够适用于求解非线性jerk方程。而且在jerk方程不含速度线性项时该方法仍然有效。     

具连续分布滞量的高阶非线性偏泛函微分方程的强迫振动性

研究一类具连续分布滞量的高阶非线性偏泛函微分方程,利用微分不等式方法,给出了这类方程在三类边值条件下解的强迫振动的充分条件,所得结果推广和包含了已知的一些结论。     

一类KdV-Burgers方程的概周期解

KdV-Burgers方程出现在许多物理模型中,是非线性科学领域中的重要模型之一.本文讨论一类具有阻尼和非齐次项的KdV-Burgers方程的概周期解存在性问题.首先利用Galerkin方法构造出方程的有界解,并利用一些数学不等式给出这个解的先验估计;然后利用所得的先验估计和标准的紧致性方法证明方程广义解的存在性;最后证明当方程的非齐次项函数是关于时间变量的概周期函数时,该广义解就是方程的概周期解.     

等截面梁纯弯曲振动的几何非线性分析

在讨论梁纯弯曲微幅振动时，考虑在材料力学的讨论中梁的挠度微分方程忽略项后，线性问题变为非线性问题。利用非线性理论对该非线性问题进行了讨论，得到了周期解稳定和不稳定区域的分界线方程和频率响应方程，得到忽略挠度几何非线性因素的条件。     

非线性扩散方程的精确解

本文研究非线性扩散方程的精确解,文中分别给出非线性扩散方程在推广旋转群和推广伸缩群上不变时满足的条件,并且利用推广旋转群和推广伸缩群构造方程的精确解,最后,给出了非线性扩散方程在推广旋转伸缩群上不变时满足的条件,推广旋转伸缩群也被用来构造非线性扩散方程的精确解.     

Dynamics of self-similar sub-10-fs-pulses in an inhomogeneous highly nonlinear fibre amplifier

ABSTRACT

We study the propagation of ultrashort pulses of width around sub-10 femtosecond in an inhomogeneous highly nonlinear single-mode fibre within the framework of a generalized higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms and spatially inhomogeneous coefficients. Additional effects to the cubic model include the distributed third-order dispersion, self-steepening, self-frequency shift due to stimulated Raman scattering, quintic nonKerr nonlinearity, derivative non-Kerr nonlinear terms, and gain or loss. The exact self-similar brightand dark-solitary-wave solutions of the governing equation are derived via a transformation connected with the constant-coefficient higher-order nonlinear Schrödinger equation with non-Kerr nonlinearity. The constraint relations among the optical fibre parameters for the existence of these self-similar structures are also discussed. Based on these exact solutions, we investigate the dynamical behaviours of self-similar localized pulses in a periodic distributed fibre system for different parameters.     

内外共振联合作用下索－梁组合结构非线性振动分析

研究内共振和外共振联合作用下的索－梁组合结构非线性振动问题。利用Ｈａｍｉｌｔｏｎ原理推导索－梁组合结构非线性动力学方程,同时考虑索的垂度以及由梁和索之间模态耦合引起的非线性影响。利用Ｇａｌｅｒｋｉｎ方法将索－梁组合结构非线性运动偏微分方程离散为一组常微分方程。最后对数学模型进行数值计算,得到了不同内外共振联合作用下梁和索的模态时程曲线。研究表明,梁的稳态运动呈现周期性振荡,而索在不同的内外共振联合作用下,分别呈现出混沌或周期性振荡,并且索和梁之间持续的模态交替现象只能在特定的内外共振下出现。     

退化环面上的非线性jerk方程近似周期解的同伦分析方法

用同伦分析法求解退化环面上的非线性Jerk方程的近似周期和近似解析周期解。所得结果表明文中得到的一阶近似周期和一阶近似解析周期解与Gottlieb用低阶谐波平衡法求解得到的结果一样。当参数和初速度较大时,一阶近似周期与精确周期的百分比误差是4.831 8%,而二阶近似周期与精确周期的百分比误差小于0.219 9%。与数值方法给出的"精确"周期解比较,二阶近似解析周期解比一阶近似解析周期解要精确的多。因此,同伦分析法是求解非线性Jerk方程的一种非常有效的方法。     

Banach空间中二阶非线性微分积分方程周期边值问题

本文通过建立一个新的比较定理，给出了Banach空间中含有导数项u’的二阶非线性微分积分方程周期边值问题最大解和最小解的存在性。     

A Refined Asymptotic Theory for the Nonlinear Analysis of Laminated Cylindrical Shells

Within the framework of the three-dimensional (3D) nonlinear elasticity, a refined asymptotic theory is developed for the nonlinear analysis of laminated circular cylindrical shells. In the present formulation, the basic equations including the nonlinear relations between the finite strains (Green strains) and displacements, the nonlinear equilibrium equations in terms of the Kirchhoff stress components and the generalized Hooke's law for a monoclinic elastic material are considered. After using proper nondimensionalization, asymptotic expansion, successive integration and then bringing the effects of transverse shear deformation into the leading-order level, we obtain recursive sets of the governing equations for various orders. It is shown that the von Karman-type first-order shear deformation theory (FSDT) is derived as a first-order approximation to the 3D nonlinear theory. The differential operators in the linear terms of governing equations for the leading order problem remain identical to those for the higher-order problems. The nonlinear terms related to the unknowns of the current order appear in a regular pattern and the other nonhomogeneous terms can be calculated by the lower-order solutions. It is also illustrated that the nonlinear analysis of laminated circular cylindrical shells can be made in a hierarchic and consistent way.     

Nonlinear zigzag theory for electrothermomechanical buckling of piezoelectric composite and sandwich plates

Summary A coupled geometrically nonlinear efficient zigzag theory is presented for electrothermomechanical analysis of hybrid piezoelectric plates. The geometric nonlinearity is included in Von Karman sense. The thermal and potential fields are approximated as piecewise linear across sublayers. The deflection accounts for the transverse normal strain due to thermal and electric fields. The inplane displacements are considered to have layerwise variations, but are expressed in terms of only five primary displacement variables, independent of the number of layers. The coupled nonlinear equations of equilibrium and the boundary conditions are derived from a variational principle. The nonlinear theory is used to obtain the initial buckling response of symmetrically laminated hybrid plates under inplane electrothermomechanical loading. Analytical solutions for buckling of simply-supported plates under thermoelectric load are obtained for comparing the results with the available exact three-dimensional (3D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 3D solution, when the pre-buckling transverse normal strain is neglected in the latter solution. The present results are also compared with the third order theory with the same number of displacement variables to highlight the positive effects of the layerwise terms in the displacement field approximations of the zigzag theory.     

一类Brusselator反应扩散模型的显式精确行波解

运用双曲函数法和吴方法,我们获得了Brusselator反应扩散模型新的显式精确行波解。这些解包括了新的奇性孤波解,同期解和有理函数解。这个方法也可用于求解其它的非线性偏微分方程。