一种适用于非均匀地形的高阶Boussinesq水波模型

推导了适用于变地形情况的高阶Boussinesq波浪模型．该模型采用自由表面边界条件作为时间步进方程,利用势函数满足的Laplace方程的解析解形式建立了自由表面边界速度和底面边界速度之间的关系,使得问题封闭．以0.5倍相对水深处的速度为基本未知量,在对Laplace方程解析解进行级数求逆时保留水深梯度的高阶项,改进了速度场的Taylor展开式．对于线性特性,进行了线性浅化和Booij反射的验证性计算．为了检验有背景流动情况下拓展的Boussinesq模型的性态,对波-流相互作用问题进行了数值模拟．数值计算结果与现有理论解或其他完全势流的数值解吻合良好,表明该模型的应用范围可以扩展到含有非均匀变化地形的问题．     

一类带强色散项DGH方程的精确解

利用改进的F-展开法,求出了一类带强色散项DGH方程的一系列类孤子解,三角函数周期解和有理数解,方程结合了KdV方程的线性色散项和C-H方程的非线性色散项.而且改进的F-展开法在借助于计算机符号系统Mathematica(Maple)下,操作方便,适用于大量的非线性偏微分方程(组),并有助于发现新解.     

狭吸积环中非轴对称动力学不稳定性的非线性演化

本文在孤子理论的框架下解释非轴对称动力学不稳定性的非线性演化.在长波不可压缩极限下,两维狭吸积环的色散关系与线性KdV方程的相同.我们认为:非线性动力学不稳定性数值模拟中的“行星状”解是KdV方程的孤子解.由于在动力学不稳定性的非线性演化中密度和熵的变化,吸积盘的涡度是不守恒量,这也使角动量在不稳定过程中重新分布.     

某些半线性椭圆方程在环域上的正对径解的存在性

利用锥拉伸与锥压缩型的Krasnosel'skii不动点定理讨论了某些二阶非线性椭圆方程在环域上关于Dirichlet边界条件的正对径解的存在性。通过考察非线性项在有界闭区间上的性质建立了若干正对径解的存在性结论。主要结论不涉及非线性项的超线性增长和次线性增长。当非线性项存在极值并满足适当条件时,主要结论是非常有效的。     

反应扩散方程在分数幂空间的整体吸引子

本文证明了具有色散的反应扩散方程在分数幂空间的整体吸引子的存在性，解决了 Ｃａｒｖａｌｈｏ Ａ．Ｎ．在文［１］中提出的问题，且去掉了文山对非线性项所附加的对称性条件     

变系数的一般型发展方程的色散估计

本文研究的是带变系数的一般型线性发展方程.首先建立了其基本解的一系列色散估计:Kato光滑型估计,极大函数估计及Strichartz估计.最后应用这些估计研究了一些非自治非线性色散方程的初值问题在H~s(R)空间中的局部可解性.     

一类含梯度项的二阶半线性椭圆型方程的爆炸解

本文运用非线性变换、摄动方法结合上、下解方法得到了一类含梯度项的二阶半线性椭圆型方程爆炸解的存在性．     

基于纵向数据的半参数变系数部分线性回归模型

纵向数据是数理统计研究中的复杂数据类型之一0,在生物、医学和经济学中具有广泛的应用．在实际中经常需要对纵向数据进行统计分析和建模．文章讨论了纵向数据下的半参数变系数部分线性回归模型,这里的纵向数据的在纵向观察在时间上可以是不均等的,也可看成是按某一随机过程来发生．所研究的半参数变系数模型包括了许多半参数模型,比如部分线性模型和变系数模型等．利用计数过程理论和局部线性回归方法,对于纵向数据下半参数变系数进行了统计推断,给出了参数分量和非参数分量的profile最小二乘估计,研究了这些估计的渐近性质,获得这些估计的相合性和渐近正态性．     

非线性动力系统的自适应显式Magnus数值方法

基于最近发展的矩阵李群上非线性微分方程的显式Magnus展式,给出了非线性动力系统的有效的数值算法,并且在数值求解过程中具有自适应的步长控制特点,可以显著地提高计算效率．最后,通过非线性动力系统典型问题Duffing方程和强刚性的Van derPol方程以及非线性振子的Hamilton方程的数值实验来说明方法的有效性．     

非线性再生散度随机效应模型的贝叶斯分析

非线性再生散度随机效应模型是一类非常广泛的统计模型，包括了线性随机效应模型、非线性随机效应模型、广义线性随机效应模型和指数族非线性随机效应模型等．本文研究非线性再生散度随机效应模型的贝叶斯分析．通过视随机效应为缺失数据以及应用结合Gibbs抽样技术和Metropolis-Hastings算法（简称MH算法）的混合算法获得了模型参数与随机效应的同时贝叶斯估计．最后，用一个模拟研究和一个实际例子说明上述算法的可行眭．     

WFQEM‐based perturbation approach and its applications in analyzing nonlinear periodic structures

Based on the weak form quadrature element method, a perturbation approach is developed. Waves propagating in periodic beams on a nonlinear elastic foundation are studied by using the new proposed method. The feasibility and accuracy of the proposed method are verified by comparing the present results with those available in literatures in linear cases. Detailed modal analysis of the linear cases is conducted in order to obtain the dispersion relations of the nonlinear cases. The theoretical results show that the dispersion relations of the nonlinear cases are amplitude dependent. Furthermore, the effects of geometric parameters and degree of nonlinearity on the amplitude‐dependent dispersion relations are discussed in detail. This work provides a new method for analyzing the dispersion relations of nonlinear periodic structures and gives some useful guidelines for designing periodic beams or pipelines with nonlinear structure–foundation interaction. Copyright © 2016 John Wiley & Sons, Ltd.     

On linear and nonlinear Rossby waves in an ocean

This paper deals with recent developments of linear and nonlinear Rossby waves in an ocean. Included are also linear Poincaré, Rossby, and Kelvin waves in an ocean. The dispersion diagrams for Poincaré, Kelvin and Rossby waves are presented. Special attention is given to the nonlinear Rossby waves on a β-plane ocean. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a modified nonlinear Schrödinger equation. The solution of this equation represents solitary waves in a dispersive medium. In other words, the envelope of the amplitude of the waves has a soliton structure and these envelope solitons propagate with the group velocity of the Rossby waves. Finally, a nonlinear analytical model is presented for long Rossby waves in a meridional channel with weak shear. A new nonlinear wave equation for the amplitude of large Rossby waves is derived in a region where fluid flows over the recirculation core. It is shown that the governing amplitude equations for the inner and outer zones are both KdV type, where weak nonlinearity is balanced by weak dispersion. In the inner zone, the nonlinear amplitude equation has a new term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude, and this term occurs to account for a nonlinearity due to the flow over the vortex core. The solution of the amplitude equations with the linear shear flow represents the solitary waves. The present study deals with the lowest mode (n=1) analysis. An extension of the higher modes (n?2) of this work will be made in a subsequent paper.     

Multiplicity of solutions for a class of impulsive differential equations with Dirichlet boundary conditions via variant fountain theorems

In this paper, we study the existence of infinitely many classical solutions for a class of second-order impulsive differential equations. By using two new fountain theorems, we deal with two cases: that when the nonlinearity is superlinear and that when it is asymptotically linear at infinity. Some recent results are extended and improved.     

An improved Serre model: Efficient simulation and comparative evaluation

The so-called Serre or Green and Naghdi equations are a well-known set of fully nonlinear and weakly dispersive equations that describe the propagation of long surface waves in shallow water. In order to extend its range of application to intermediate water depths, some modifications have been proposed in the literature. In this work, we analyze a new Serre model with improved linear dispersion characteristics. This new Serre system, herein denoted by Serre_{α, β}, presents additional terms of dispersive origin, thus extending its applicability to more general depth to wavelength ratios.A careful development of the Serre_{α, β} model allows a straightforward and efficient numerical implementation. This model is suitable for numerical integration by a splitting strategy which requires the solution of a hyperbolic problem and a dispersive problem. The hyperbolic part is discretized using a high-order finite volume method. For the dispersive part standard finite differences are used. A set of numerical experiments are conducted to validate the Serre_{α, β} model and to test the robustness of our numerical scheme. Theoretical solutions and benchmark experimental data are used. Moreover, comparisons against the classical Serre equations and against another well established Serre model with improved dispersion characteristics are also made.     

Optical soliton perturbation with time-dependent coefficients in a log law media

This paper obtains the exact 1-soliton solution to the nonlinear Schrödinger’s equation with log law nonlinearity in presence of time-dependent perturbations. The dispersion and nonlinearity are also taken to be time-dependent. The perturbation terms that are considered are linear attenuation and inter-modal dispersion. The constraint condition between the time-dependent coefficients also fall out as a necessary condition for the solitons to exist.     

An iterative langevin solution for turbulent dispersion in the atmosphere

In this work we present an alternative hybrid method to solve the Langevin equation and we apply it to simulate air pollution dispersion in inhomogeneous turbulence conditions. The method solves the Langevin equation, in semi-analytical manner, by the method of successive approximations or Picard's Iterative Method. Solutions for Gaussian and non-Gaussian turbulence conditions, considering Gaussian, bi-Gaussian and Gram–Charlier probability density functions are obtained. The models are applied to study the pollutant dispersion in all atmospheric stability and in low-wind speed condition. The proposed approach is evaluated through the comparison with experimental data and results from other different dispersion models. A statistical analysis reveals that the model simulates very well the experimental data and presents results comparable or even better than ones obtained by the other models.     

Analytical study for the ability of nonlinear transmission lines to generate solitons

The ability of the nonlinear transmission lines (NLTL) has been studied analytically, in this paper to generate solitons and to cause waveform spreading. This can be achieved by balancing nonlinearity and dispersion. A new technique of improved tanh method (ITM) and improved sech methods (ISM) is applied to the nonlinear partial differential equation that describes the NLTL. It is found that the parameters of the transmission line play an important role in controlling the shape of the soliton.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

A nonlinear boundary eigenvalue problem for TM-polarized electromagnetic waves in a nonlinear layer

We consider the propagation of TM-polarized electromagnetic waves in a nonlinear dielectric layer located between two linear media. The nonlinearity in the layer is described by the Kerr law. We reduce the problem to a nonlinear boundary eigenvalue problem for a system of ordinary differential equations. We obtain a dispersion relation and a first approximation for eigenvalues of the problem. We compare the results with those obtained for the case of a linear medium in the layer.     

Propagation of TM waves in a Kerr nonlinear layer

TM electromagnetic waves propagating through a nonlinear homogeneous isotropic unmagnetized dielectric layer located between two homogeneous isotropic half-spaces are studied. The nonlinearity in the layer obeys the Kerr law. The problem is reduced to a system of nonlinear ordinary differential equations. A dispersion relation for the propagation constants is derived. The results are compared with those in the case of a linear layer.