缓变任意截面渠道中的非线性周期波以及孤立波的分裂

本文研究了在流动方向有缓慢变化的任意截面渠道中的非线性周期波、孤立波以及孤立波在这种渠道中的分裂;导出了适用于这种渠道的变系数KdV方程,并求出了该方程的首项近似解;得出了波速、周期、波高和渠道几何尺寸之间的关系,得到了分裂后孤立波个数的判别式及分裂后孤立波波幅的表示式,并应用于矩形渠道和左右对称的三角形渠道。对于矩形渠道的情况,本文的结果和Madsen和Mei,Johnson,Svendsen和Buhr Hansen等人的结果一致。     

流动对孤立波演变的影响

本文研究了流动对二维缓变渠道中的孤立波的演变的影响,导出了远场理论的基本方程——变系数KdV方程,得到了孤立波解。文章还讨论了孤立波的分裂,找出了Q～h平面上的分裂区域及分裂后孤立波数目的判别式。     

组合KdV方程的孤立波解与相似解

本文讨论组合KdV方程孤立波解的一个性质,指出该方程可化为Painlevé方程,并利用相似变量的特殊变换导出一类新的偏微分方程.     

一类5阶KdV方程的孤立波解

近年来,关于3阶KdV方程的孤立波解得到迅速发展,而对于5阶KdV方程的孤立波解文献报道较少.本文主要采用Sine-Cosine展开法得到了一类5阶KdV方程的孤立波解;然后利用Matlab计算软件,获得了孤立波解的图形,其结果展示了孤立子与系数之间的相互关系;最后,应用所得的结果分别得到了Lax方程, SK方程, CDG方程的孤立波解.     

缓变深度分层流体中的准周期波和准孤立波

本文讨论具缓变深度二流体系统中的非线性波,该系统由一不规则底部与一水平固壁间的两层常密度无粘流体所组成.文中用约化摄动法导出了所考虑模型的变系数Korteweg-de Vries方程,并用多重尺度法求出了该方程的近似解,发现底部固壁的不规则变化将产生所谓准周期波和准孤立波.它们的周期、波速和波形将发生缓慢变化,文中给出了准周期波的周期随深度的变化关系式以及准孤立波波幅、波速随深度的变化关系式,底部水平情形和单层流体情形可看成是本文的特例.     

三层流体系统中孤立波的产生

采用约化摄动方法导出了三层流体系统中各界面所遵循的KdV方程,讨论了流体深度对孤立波产生的影响,将波分为快模式、中间模式和慢模式波后发现慢模式波的结果与已有的实验结果定性上一致.此外,还发现自由面上可能存在下凹的孤立波,这有待于实验的验证.     

孤立Rossby波的fmKdV方程及其解析解（英文）

本文基于刻画非线性Rossby波浅水模式的准地转位涡方程,利用约化摄动法,推导了强迫的修正Korteweg-de Vries(fm Kd V)方程.通过分析fm Kd V方程,得到基本地形是Rossby孤立波形成的主要因素,缓变地形是外强迫因素.除此之外,推广beta效应和基本剪切流也是孤立波形成重要因素.最后,利用广义形变映射法获得fm Kd V方程的解析解,结果表明缓变地形影响孤立波的速度,而耗散影响孤立波的速度和振幅.     

有限变形弹性杆中三种非线性弥散波

在一维弹性细杆拉压、扭转和弯曲波的经典线性理论基础上,分别计入有限变形和弥散效应,借助Hamilton变分原理,由统一的方法导出了3种非线性弥散波的演化方程．对3种演化方程进行了定性分析．结果表明,这些方程在相平面上存在同宿轨道或异宿轨道,分别相应于孤波解或冲击波解．根据齐次平衡原理,用Jacobi椭圆函数展开对这些演化方程进行了求解,在一定的条件下它们均可能存在孤立波解或冲击波解,这与方程的定性分析完全一致．     

Klein-Gordon方程的定性分析及精确解

本文运用动力系统的定性理论,得到了Klein-Gordon方程的孤立波分支图,并求得了与定性分析结果一致的孤波解.     

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。     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

On the solitary wave dynamics,under slowly varying medium,for nonlinear Schrödinger equations

We consider the problem of a solitary wave propagation, in a slowly varying medium, for a variable-coefficients nonlinear Schrödinger equation. We prove global existence and uniqueness of solitary wave solutions for a large class of slowly varying media. Moreover, we describe for all time the behavior of these solutions, which include refracted and reflected solitary waves, depending on the initial energy.     

Solitary Waves and Their Linear Stability in Nonlinear Lattices

Solitary waves in a general nonlinear lattice are discussed, employing as a model the nonlinear Schrödinger equation with a spatially periodic nonlinear coefficient. An asymptotic theory is developed for long solitary waves, which span a large number of lattice periods. In this limit, the allowed positions of solitary waves relative to the lattice, as well as their linear stability properties, hinge upon a certain recurrence relation which contains information beyond all orders of the usual two‐scale perturbation expansion. It follows that only two such positions are permissible, and of those two solitary waves, one is linearly stable and the other unstable. For a cosine lattice, in particular, the two possible solitary waves are centered at a maximum or minimum of the lattice, with the former being stable, and the analytical predictions for the associated linear stability eigenvalues are in excellent agreement with numerical results. Furthermore, a countable set of multi‐solitary‐wave bound states are constructed analytically. In spite of rather different physical settings, the exponential asymptotics approach followed here is strikingly similar to that taken in earlier studies of solitary wavepackets involving a periodic carrier and a slowly varying envelope, which underscores the general value of this procedure for treating multiscale solitary‐wave problems.     

Do peaked solitary water waves indeed exist?

Many models of shallow water waves, such as the famous Camassa–Holm equation, admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa–Holm equation. Besides, it is proved that Kelvin’s theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, unlike the traditional smooth waves that are dispersive with wave height, the phase speed of the peaked solitary waves has nothing to do with wave height, but depends (for a fixed wave height) on its decay length, i.e., the actual wavelength: in fact, the peaked solitary waves are dispersive with the actual wavelength when wave height is fixed. In addition, unlike traditional smooth waves whose kinetic energy decays exponentially from free surface to bottom, the kinetic energy of the peaked solitary waves either increases or almost keeps the same. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.     

Model Equations for Gravity-Capillary Waves in Deep Water

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gravity-capillary dispersion relation, but preserving exactly either a quadratic energy or a momentum. Solitary wavepacket waves are calculated for each model. Each model supports the elevation and depression waves known to exist in the Euler equations. The stability of these waves is discussed, as is the dynamics resulting from instabilities and solitary wave collisions.     

Effect of Deborah number and phase difference on peristaltic transport of a third-order fluid in an asymmetric channel

The effect of a third-order fluid on the peristaltic transport in an asymmetric channel is studied. The wavelength of the peristaltic waves is assumed to be large compared to the varying channel width, whereas the wave amplitudes need not be small compared to the varying channel width. The channel asymmetry is produced by choosing the peristaltic wave train on the walls to have different amplitudes and phase. The flow is investigated in a wave frame of reference moving with velocity of the wave. The effects of Deborah number, phase difference, varying channel width and wave amplitudes on the pumping characteristics, streamline pattern and trapping phenomena are investigated. It is observed that the trapping regions increase as the channel becomes more and more symmetric and the trapped bolus volume decreases for increasing Deborah number, phase difference and varying channel width whereas it increases for increasing flow rate and wave amplitudes. Furthermore, the obtained results could also have applications to a range of peristaltic flows for a variety of non-Newtonian fluids such as aqueous solutions of high-molecular weight polyethylene oxide and polyacrylamide.     

耦合BBM方程组孤立波解的轨道稳定性(英文)

本文研究具有Hamilton形式的耦合BBM方程组孤立波解的轨道稳定性.首先找到两族显式孤立波解.然后通过详细的谱分析证明出孤立波解的轨道稳定性.     

Construction of traveling wave solutions of the (2 + 1)-dimensional modified KdV-KP equation

Traveling wave solutions have played a vital role in demonstrating the wave character of nonlinear problems emerging in the field of mathematical sciences and engineering. To depict the nature of propagation of the nonlinear waves in nature, a range of nonlinear evolution equations has been proposed and investigated in the existing literature. In this article, solitary and traveling periodic wave solutions for the (2 + 1)-dimensional modified KdV-KP equation are derived by employing an ansatz method, named the enhanced (G′/G)-expansion method. For this continued equation, abundant solitary wave solutions and nonlinear periodic wave solutions, along with some free parameters, are obtained. We have derived the exact expressions for the solitary waves that arise in the continuum-modified KdV-KP model. We study the significance of parameters numerically that arise in the obtained solutions. These parameters play an important role in the physical structure and propagation directions of the wave that characterizes the wave pattern. We discuss the relation between velocity and parameters and illustrate them graphically. Our numerical analysis suggests that the taller solitons are narrower than shorter waves and can travel faster. In addition, graphical representations of some obtained solutions along with their contour plot and wave train profiles are presented. The speed, as well as the profile of these solitary waves, is highly sensitive to the free parameters. Our results establish that the continuum-modified KdV-KP system supports solitary waves having different shapes and speeds for different values of the parameters.     

Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel

An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.     

Solitary wave solutions for a general Boussinesq type fluid model

The possible solitary wave solutions for a general Boussinesq (GBQ) type fluid model are studied analytically. After proving the non-Painlevé integrability of the model, the first type of exact explicit travelling solitary wave with a special velocity selection is found by the truncated Painlevé expansion. The general solitary waves with different travelling velocities can be studied by casting the problems to the Newtonian quasi-particles moving in some proper one dimensional potential fields. For some special velocity selections, the solitary waves possess different shapes, say, the left moving solitary waves may possess different shapes and/or amplitudes with those of the right moving solitons. For some other velocities, the solitary waves are completely prohibited. There are three types of GBQ systems (GBQSs) according to the different selections of the model parameters. For the first type of GBQS, both the faster moving and lower moving solitary waves allowed but the solitary waves with“middle” velocities are prohibit. For the second type of GBQS all the slower moving solitary waves are completely prohibit while for the third type of GBQS only the slower moving solitary waves are allowed. Only the solitary waves with the almost unit velocities meet the weak non-linearity conditions.