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.     

一类描述大气中重力孤立波的新模型

从两层流体浅水波方程出发,运用尺度分析与扰动方法,建立了一类新的模型(mKdV-BO模型)来描述大气中的重力孤立波。前人建立的KdV模型和BO模型适合描述经向和纬向扰动较弱时重力孤立波的生成和演化,而该模型的非线性更强,适合描述经向、纬向扰动较强时重力孤立波的生成与演化。通过运用试探函数法获得了模型的代数孤波解,并分析了孤立波的生成条件与传播速度。新模型的建立对于进一步解释大气中列队雷雨阵的形成机制,探讨大气中的强对流天气如飑线的形成等具有重要意义。 关键词：重力孤立波;试探函数法;列队雷雨阵     

表面张力对非传播孤立波的影响

本文在研究非传播弧立波时仔细考虑了表面张力的影响,把表面张力和液体深度的参数平面划分为三个区域,发现其中两个区可产生呼吸弧立波。到目前为止,所有理论和实验文章中提到的呼吸弧立波的参数都在一个参数区内,我们首先报道了另一个参数区并被我们的实验证实.在第三个参数区中,理论分析得到的解是纽结孤立波,但是在我们的实验中除了得到纽结孤立波之外,过得到了一种类似于呼吸孤立波的非传播孤立波.     

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.     

立方非线性微结构固体中的对称孤立波及存在条件

考虑固体材料的宏观尺度立方非线性效应、微尺度立方非线性效应以及微尺度频散效应并根据修正的Mindlin理论,建立了一维微结构固体中纵波传播的一种新模型.用动力系统的定性分析方法,证明了适当条件下立方非线性微结构固体中可存在对称钟型孤立波和反钟型孤立波,并给出了两种孤立波的存在条件.用数值方法分析了微尺度立方非线性效应对钟型与反钟型孤立波的影响,结果显示随着微尺度非线性效应的增强(或负增强),两种孤立波的宽度变窄(或变宽)而幅度保持不变.     

深分层流体中内孤立波的相互作用——Ⅰ.弱相互作用 *

采用Lagrange观点研究分层流体中内孤立波的弱相互作用,它包括不同模式孤立波间的追撞和迎撞,以及相同模式孤立波间的迎撞.分析表明在有限深度情形每个波遵循ILW方程,而在无限深度情形每个波满足Benjamin-Ono方程,相互作用的主要效应体现在相移上.     

自洽磁场作用下的等离子体中的孤立波

本文发展了文献[1]中的方法,提出了一个直接从全波段色散关系判别可能存在孤立波的波段的方法——色散关系法;并用它证明了平行或垂直于自洽磁场传播的10种线性波中,只有磁声波和电子哨波可能存在与之对应的孤立波。     

参数激励下的内波孤立子和扭结 *

在不相容、不等密度的双层液体的参数激励下的Faraday实验中,观察到了非传播界面波孤立子和扭结以及稳定的双孤立子等现象.总体上看,界面波现象和已有的表面波现象基本上一致,只是由于上层液体的存在,使得界面波的模式振动频率明显红移、波形变矮变宽、其稳定性也不如表面波.在理论上,对流体力学方程组及其相应的边界和界面条件进行了约化,得到了一个有阻尼、带驱动的非线性Schr(?)dinger方程.从而,令人满意地同时解释了界面孤立子波和扭结波.无论从实验现象上,还是从理论上看,自由表面波只是界面波的一个特例.     

一个两流体系统中mKdV孤立波的迎撞*

本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化.     

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.     

The Korteweg-de Vries equation in a cylindrical pipe

A mathematical model of nonlinear wave propagation in a pipeline is constructed. The Korteweg-de Vries equation is derived by determining asymptotic solutions and changing variables. A particular solution to the model equations is found that has the fluid velocity function in the form of a solitary wave. Thus, the class of nonlinear fluid dynamics problems described by the KdV equation is expanded.     

Explicit solutions of the Camassa–Holm equation

Explicit travelling-wave solutions of the Camassa–Holm equation are sought. The solutions are characterized by two parameters. For propagation in the positive x-direction, both periodic and solitary smooth-hump, peakon, cuspon and inverted-cuspon waves are found. For propagation in the negative x-direction, there are solutions which are just the mirror image in the x-axis of the aforementioned solutions. Some composite wave solutions of the Degasperis–Procesi equation are given in an appendix.     

Stability of the Camassa-Holm solitons

Summary. We consider the stability problem of the solitary wave solutions of a completely integrable equation that arises as a model for the unidirectional propagation of shallow water waves. We prove that the solitary waves possess the spectral properties of solitons and that their shapes are stable under small disturbances.     

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.     

Painlevé property, auto-Bäcklund transformation and analytic solutions of a variable-coefficient modified Korteweg-de Vries model in a hot magnetized dusty plasma with charge fluctuations

Under investigation in this paper is a variable-coefficient modified Korteweg-de Vries (vc-mKdV) model in a hot magnetized dusty plasma with charge fluctuations. With symbolic computation and bilinear method, Painlevé property is studied, auto-Bäcklund transformation is constructed, while soliton and other analytic solutions are obtained. Furthermore, influence of the coefficients on the dust-ion-acoustic (DIA) solitary wave propagation is investigated based on the soliton solution, which can be concluded as follows: (i) Amplitude of the DIA solitary wave is proportional to the square of the ratio of the coefficients of the dispersive to nonlinear terms; (ii) Velocity of the DIA solitary wave is controlled by the coefficients of the dispersive and dissipative terms; (iii) Propagation trajectory of the DIA solitary wave depends on the function forms of the coefficients of the dispersive, nonlinear and dissipative terms.     

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.     

Universally-Convergent Squared-Operator Iteration Methods for Solitary Waves in General Nonlinear Wave Equations

Three new iteration methods, namely the squared-operator method, the modified squared-operator method, and the power-conserving squared-operator method, for solitary waves in general scalar and vector nonlinear wave equations are proposed. These methods are based on iterating new differential equations whose linearization operators are squares of those for the original equations, together with acceleration techniques. The first two methods keep the propagation constants fixed, while the third method keeps the powers (or other arbitrary functionals) of the solution fixed. It is proved that all these methods are guaranteed to converge to any solitary wave (either ground state or not) as long as the initial condition is sufficiently close to the corresponding exact solution, and the time step in the iteration schemes is below a certain threshold value. Furthermore, these schemes are fast-converging, highly accurate, and easy to implement. If the solitary wave exists only at isolated propagation constant values, the corresponding squared-operator methods are developed as well. These methods are applied to various solitary wave problems of physical interest, such as higher-gap vortex solitons in the two-dimensional nonlinear Schrödinger equations with periodic potentials, and isolated solitons in Ginzburg–Landau equations, and some new types of solitary wave solutions are obtained. It is also demonstrated that the modified squared-operator method delivers the best performance among the methods proposed in this article.