低纬地区非线性孤立波存在性讨论

本文讨论了静力平衡下的绝热自由大气非线性重力惯性波的孤立渡解存在性,设运动在y方向是均匀的,又采用了β平面近似,同时忽略了动量方程中垂直平流项,我们导出了孤立波解的解析解表达式,通过对波速c的讨论,我们指出:在中纬地区,很难形成孤立波;在低纬地区(热带地区),由于科氏力很小,层结稳定度较弱有可能产生孤立波。最后我们讨论了孤立波的波速c与科氏力f,β和稳定参数σ_8之间关系。     

旋转层结流体中垂直切变基本流、β效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法(多尺度方法),从位涡守恒方程出发,分析旋转层结大气中基本流有垂直切变以及层结效应对β效应、地形效应和强迫耗散共同作用下的Rossby波的影响,得到一个非标准形式的非线性Schr?dinger方程,而在水平波数小于3时该方程有包络孤立波解;又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外,本文还应用常数变异法求解了非齐次的Bessel方程,得到包络Rossby孤立波的经向结构.     

旋转层结流体中垂直切变基本流、$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波

本文运用摄动法和WKB方法（多尺度方法）, 从位涡守恒方程出发, 分析旋转层结大气中基本流有垂直切变以及层结效应对$\beta$效应、地形效应和强迫耗散共同作用下的Rossby波的影响, 得到一个非标准形式的非线性Schr\"{o}dinger方程,而在水平波数小于3时该方程有包络孤立波解; 又进一步说明基本流的垂直切变对包络Rossby孤立波的波速的影响;强迫耗散对包络Rossby孤立波稳定度的影响.另外, 本 文还应用常数变异法求解了非齐次的Bessel方程, 得到包络Rossby孤立波的经向结构.     

基流风速包含垂直变化时的三维无旋Non-Boussinesq流的地形重力波拖曳解

用WKB近似方法建立了表达三维地形重力波拖曳的解析Non-Boussinesq扰动模型,其中在大Richardson数条件下给出了(静力和非静力模型的)重力波拖曳及其地表扰动气压的二阶表达式.通过针对经典的理想化三维圆钟型山体的一个算例证明,当基流风速切变为线性时,重力波拖曳随着切变的增强而减弱;并且前向垂直切变(forward-shear,风速随高度增加)所对应的重力波拖曳比反向切变(backward-shear,风速随高度减小)所对应的重力波拖曳减弱得更快.这种现象与模型是否采用静力近似无关.     

非传播孤立波及其相互作用的数值模拟

通过数值求解由Miles导出的目前公认的的非传播孤立波的控制方程——一个带复共轭项的非线性立方SchrLdinger方程,对非传播孤立波进行研究。讨论了Miles方程中的线性阻尼系数α的值,计算表明,线性阻尼α对形成稳定的非传播孤立波影响很大,Laedke等人关于非传播孤立波的稳定性条件只是一个必要条件,而不是充分条件。模拟了两个非传播孤立波的相互作用,数值模拟表明,两个波的作用模式依赖于系统的参数,对不同的初始扰动及其演化的计算表明,只有适当的初始扰动才能形成单个稳定的非传播孤立波,否则扰动可能消失或发展成多个孤立波。     

微结构固体中孤立波的演变及非光滑孤立波

给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.     

完整Coriolis力作用下的非线性近惯性波

从包含有完整Coriolis力作用下的大气运动原始基本方程组出发,通过尺度分析,采用多重尺度法及摄动展开法,推导了中高纬大气非线性近惯性波振幅演化所满足的Korteweg-de Vries方程.从演化方程的结果可以看出Coriolis参数水平分量对非线性近惯性波的影响,主要体现为对频散效应的修正及与基本流的相互作用.从理论上解释了完整Coriolis力作用下的中高纬地区大气非线性近惯性波运动的物理机制.     

光化-动力耦合重力波模式及其应用——Ⅰ. 模式的建立

利用数值模拟的方法研究了大气重力内波传播的扰动对中间层上部和低热层大气化学成分分布的影响. 着重探讨非线性、可压缩的二维光化-动力耦合重力波模式的建立、计算方法以及模式正确性分析. 该模式考虑了光化学反应产生的非绝热效应以及重力波对大气化学反应和对化学成分输运的影响. 模式在水平方向利用拟谱方法,对垂直方向和时间采用有限差分方法, 并利用FICE方法对模式求解. 对小振幅波动的模拟结果与线性重力波理论结果符合得很好, 证明了该模式算法的正确性, 它表明拟谱法和有限差分法相结合是大气重力波模拟研究的有效的、并且计算量较小的方法之一.     

孤立子型热源强迫与大气和海洋流场的奇异响应

运用非线性动力学分析方法,对一个广义的地球流体动力学正压准地转模式中,局地热源强迫对大气和海洋流场产生的响应结果,进行了解析研究.发现热源扰动和响应流场之间存在很好的对应关系:两者具有类似的分布模态,且孤立子型的响应流场要比孤立子型的热源强迫范围大得多,即"狭窄"的热源扰动可能导致"宽广"的流场响应;强迫热源的特殊结构有可能导致大气或海洋流场出现奇异响应,从而使大气或海洋环流发生异常(如大气阻塞现象);中、高纬地区的大气和海洋流场对热源强迫的响应要比低纬地区显著得多.上述结果与中、低纬大气试验和观测资料的研究结果相符,可部分解释地球流体中,局地热源异常可能导致的环流异常现象.     

全球大气重力波联测期间电离层扰动的台阵观测

本文利用武汉电离层观象台的加密频高图和三站Doppler图记录及新近提出的短波扰动反演方法,分析了1985年全球大气重力波联测(WAGS)期间10月18日这天白天的重力波扰动,发现在该日扰动中有两列不同形态的波列分别属于中尺度的内重力波和大尺度的导制重力波。本文还进一步估算了这两列重力波的水平传播参量,深入分析了它们的频谱结构的高度变化特征,发现了重力波功率谱谱峰的分裂和偏移等重要现象。本文结果表明,新的反演方法使电离层无线电汉诊断的简易短波实验系统,也能成为探测和研究重力波一类大尺度电离层动力过程的有效手段。     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

基于吴方法的孤波自动求解软件包及其应用

基于非线性代数方程组的吴特征列方法,在计算机代数系统Maple上实现了非线性微分方程孤波解的自动求解,编制了一个小型实用的软件包。作为应用,考虑了一个一般的五阶模型方程,利用该软件包获得了此方程新的孤波解以及孤子解存在的条件。     

Perturbation Theory for the Solitary Wave Solutions to a Sasa‐Satsuma Model Describing Nonlinear Internal Waves in a Continuously Stratified Fluid

The adiabatic evolution of perturbed solitary wave solutions to an extended Sasa‐Satsuma (or vector‐valued modified Korteweg–de Vries) model governing nonlinear internal gravity propagation in a continuously stratified fluid is considered. The transport equations describing the evolution of the solitary wave parameters are determined by a direct multiple‐scale asymptotic expansion and independently by phase‐averaged conservation relations for an arbitrary perturbation. As an example, the adiabatic evolution associated with a dissipative perturbation is explicitly determined. Unlike the case with the dissipatively perturbed modified Korteweg–de Vries equation, the adiabatic asymptotic expansion for the Sasa‐Satsuma model considered here is not exponentially nonuniform and no shelf region emerges in the lee‐side of the propagating solitary wave.     

二维浅水波方程的间断周期解和间断孤立波解讨论

本文讨论地球流体运动浅水波模式中间断周期解与间断孤立波解.在系统的非平衡点即奇点附近考虑轨线性质时,我们发现只要引入广义解的概念(分片光滑连续解),就会产生间断周期解并得到了间断周期解的条件.当系统在退化的过程中,发现系统此时会产生间断的孤立波解,与此同时其它物理量也产生了间断.这里我们发现,一般认为在超高速情况下解会产生间断,然而在非超高速时也会产生间断现象.本文讨论了上述一系列问题得到了间断解的解析解表达式,并把这一事实与飑线的实例进行比较,得到了不少类似之处.     

Regularity and symmetry properties of rotational solitary water waves

We prove that a solitary water wave driven by gravity has real-analytic streamlines for arbitrary vorticity functions if the flow contains no stagnation points. Based on this property, we show that if all the streamlines attain their global maximum (resp. minimum) on the same vertical line, then the solitary wave has to be symmetric and strictly monotone away from the crest (resp. trough). Our results are true for sub- and supercritical solitary waves as well.     

Conservation Laws,Hamiltonian Structure,Modulational Instability Properties and Solitary Wave Solutions for a Higher‐Order Model Describing Nonlinear Internal Waves

Recent theoretical advances in connecting the wave‐induced mean flow with the conserved pseudomomentum per unit mass has permitted the first rational derivation of a model that describes the weakly nonlinear propagation of internal gravity plane waves in a continuously stratified fluid. Depending on the particular parameter regime examined the new model corresponds to an extended bright or dark derivative nonlinear Schrödinger equation or an extended complex‐valued modified Korteweg‐de Vries or Sasa–Satsuma equation. Mass, momentum, and energy conservation laws are derived. A noncanonical infinite‐dimensional Hamiltonian formulation of the model is introduced. The modulational stability characteristics associated with the Stokes wave solution of the model are described. The bright and dark solitary wave solutions of the model are obtained.     

Irrotational solitary waves with surface-tension

Purely capillary solitary waves cannot be obtained under the systematic shallow water theory developed by Friedrichs. In fact, if we neglect the gravity and take into account the surface-tension only at the free surface and proceed on the lines of Keller, who obtained cnoidal and solitary waves using the Friedrichs’ shallow water systematic theory, we get nothing other than uniform flow. In this article, on the lines of Friedrichs and Hyers, we find the solitary wave motion when surface-tension is also taken into account along with the gravity andgh/U² < 1, whereg is the acceleration due to gravity,h and U are the depth and the horizontal velocity of the liquid at infinity. The solution is sought in the form of infinite series in ascending powers of a suitably defined parameter after giving a stretching in the horizontal direction.     

Explicit exact solutions for compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order

In this paper, we improved a method presented previously (Phys. Lett. A 285 (2001) 355) by means of a proper transformation. Applying the improved method, we consider the generalized compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions, are obtained.     

Some new nonlinear wave solutions and their convergence for the (2+1)‐dimensional Broer–Kau–Kupershmidt equation

We use the bifurcation method of dynamical systems to study the (2+1)‐dimensional Broer–Kau–Kupershmidt equation. We obtain some new nonlinear wave solutions, which contain solitary wave solutions, blow‐up wave solutions, periodic smooth wave solutions, periodic blow‐up wave solutions, and kink wave solutions. When the initial value vary, we also show the convergence of certain solutions, such as the solitary wave solutions converge to the kink wave solutions and the periodic blow‐up wave solutions converge to the solitary wave solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of Exp-function method to generalized Burgers-Fisher equation

In this paper,the Exp-function method is used to construct exact solitary wave solutions for the generalized Burgers-Fisher equation with nonlinear terms of any order.With the aid of Maple computation,we obtain many new and more general exact solitary wave solutions expressed by various exponential and hyperbolic functions.Our results can successfully recover previously known solitary wave solutions that have been found by the tanh-function method and other more sophisticated methods.