带强迫项的变系数KdV方程的多孤立波解及其应用

本文利用改进的齐次平衡法,首先得到了带强迫项的变系数KdV方程的多孤立波解,然后借助此解得到了强迫KdV方程的多孤立波解.最后作为应用例子,利用图形分析方法分析了Rossby孤立波的相互作用,指出了影响Rossby孤立波相对幅度、相位、传播方向及平衡位置的主要原因.     

含微结构二维固体中非对称孤立波的存在条件

根据Mindlin微结构理论重新推导了含微结构的二维固体中孤立波传播的控制方程.利用行波变换,把复杂的非线性偏微分方程组简化为一非线性常微分方程.最后用动力系统定性分析理论,分析了含微结构的二维固体中孤立波的存在条件及其几何特性,证明了当介质中的某些参数满足适当条件时,在含微结构的二维固体中可以存在一种非对称孤立波.     

微结构固体中孤立波的动力稳定性

描述微结构固体中波传播的一种KdV类方程作为控制方程并利用积分因子方法,对微结构固体中传播孤立波的动力学稳定性进行了数值模拟研究.主要以高斯波、Ricker子波以及双曲正割波扰动作为初始扰动,考察了不同小扰动下孤立波能否较长时间保持波形结构和传播速度而稳定传播问题.结果表明,不同的小扰动对孤立波的影响不同,孤立波的稳定传播与扰动幅度和宽度都有关系,只有受到幅度和宽度都非常小的扰动下在弱微尺度非线性效应的微结构固体中传播的孤立波才能显现出一定程度的抗干扰性和动力学稳定性,能够在微结构固体中较长时间稳定传播.     

具强迫项的变系数Burgers方程的特殊孤波结构

利用推广的双曲函数展开法,得到了具强迫项的变系数Burgers方程的几组带有任意函数和任意常数的精确解.根据得到的解,分析了各种可能的孤波结构,发现了运动学特征不同于通常扭结孤立波的特殊扭结孤立波.     

微结构固体中孤立波的动力学稳定性

描述微结构固体中波传播的一种KdV类方程作为控制方程并利用积分因子方法,对微结构固体中传播的孤立波的动力学稳定性进行了数值模拟研究。主要以高斯波、Ricker子波以及双曲正割波作为初始扰动,考察了不同小扰动下孤立波能否较长时间保持波形结构和传播速度而稳定传播问题。模拟结果表明,不同的小扰动对孤立波的影响不同,孤立波的稳定传播与扰动幅度和宽度都有关系,只有受到幅度和宽度都非常小的扰动下在微结构固体中传播的孤立波才能显现出一定程度的抗干扰性和动力学稳定性,可在微结构固体中较长时间稳定传播。     

Burgers方程的精确孤立波解的符号计算

借助于符号计算Maple,给出了一种构造非线性波动方程行波解的直接代数方法,该方法的主要特点是充分利用Riccati方程.使用此方法得到Burgers方程的多组精确行波解,其中包括一些新的孤立波解,这种方法也适用于求解其它的非线性波动方程（组）.     

基于波环粒子的实时水波仿真方法

为了降低水波模拟过程中的计算成本并提高其扩散现象的逼真度，提出一种基于波环粒子包的实时二维平面水波仿真方法。该方法采用波环粒子为基本计算单元，粒子内部继承“波包”的概念，使用多个频段水波叠加的方式再现水波视觉效果。在计算水波反射过程时，通过添加镜像波源的形式减少碰撞计算，避免复杂几何判定。为适应不同硬件的计算性能差异，该方法提供额外的计算精度参数，可针对不同硬件计算能力调节水波反射计算复杂度。实验结果表明，该方法可使用较少的粒子模拟出较为真实的水波运动，且避免了碰撞反射后水波断裂的问题。在相同硬件平台上的性能测试显示，所提波环仿真方法的渲染帧率比传统波包算法高出至少60%，在一些水波状态特别复杂的情况下可达到400%以上的加速效果。     

改进同伦分析方法及推广Kuramoto Sivashinsky方程的近似解

首先介绍了带有两个辅助参数的改进同伦分析方法,然后用该方法得到了推广Kuramoto-Sivashin-sky方程的同伦近似解.所得近似解与精确孤立波解进行比较,发现本文得到的近似解更有效地逼近真实解.因为该解包含了两个辅助参数,所以能够更有效地调节和控制其收敛区域和速度.研究表明带有两个辅助参数的改进同伦分析方法对复杂非线性系统的研究更有它的优点.     

应用SPH方法实现海面孤立波运动的模拟

首先介绍孤立波的Kd V方程,继而讨论了孤立波SPH方法的数值求解过程,选择SPH光滑核函数作为正则化高斯核函数。分析了数值求解过程的时间积分方法,给出了具体计算公式,最后给出相应程序中的具体参数下孤立波运动模拟效果。     

对称正则长水波方程的畸形波解（英文）

众所周知,非线性发展方程在准确描述自然现象方面起着很重要的作用。我们已经应用许多方法获得了孤子解和周期波解。本论文中,将发现非线性发展方程的一种叫做畸形波解的新类型的解,同时也提出了寻求方程畸形波解的一种新的方法 HBLM。(1+1)维对称正则长水波方程(SRLW)作为本文的案例,强调了所提方法的有效性。     

Exact periodic,cusp, solitary and loop wave solutions of the EX-ROE

The extended reduced Ostrovsky equation (EX-ROE) is investigated by using the dynamical system theory. The bifurcation phase portraits are drawn in different regions of parameter plane. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakon, solitary loop waves and periodic loop waves are obtained. The dynamic characters of these solutions are investigated.     

Solitary wave families of NLPDES via reversible systems theory

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including on other NLPDEs, are also mentioned.     

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

Estimating parameters of a two-layer stratified ocean from polarity conversion of internal solitary waves observed in satellite SAR images

This paper presents a method for estimating parameters of a two-layer stratified ocean using satellite SAR images. According to weak nonlinearity shallow water theory, internal solitary waves (ISWs) in stratified oceans may be either depression or elevation waves, depending on the sign of the quadratic nonlinearity coefficient in the KdV equation. It has been confirmed that ISWs can convert their polarity when passing through a turning point, where the quadratic nonlinearity coefficient changes sign. For a two-layer stratified ocean, the turning point is located where the upper and lower layer depths are equal. The authors suggest that depression, elevation and broadening ISWs can be discerned according to their different signatures in SAR images. It is also found that a SAR image can record a continuous evolution process from depression to elevation ISWs in its spatial domain, under conditions of a spatially inhomogeneous ocean environment. Therefore, the upper and lower layer depths can be calculated by determining the polarity conversion of ISWs observed in satellite SAR images. Furthermore, the density difference between the upper and lower layers can also be estimated, when the wave speed is known. We extract ocean stratification parameters, including upper layer depth and density difference, from polarity conversion of ISWs observed in a RADARSAT-1 SAR image taken over the northeastern South China Sea. Comparing the estimated results with field measurements, we find that this method can estimate the upper layer depth with considerable success. In estimating the density difference between the upper and lower layers, it also gives a quite reasonable result.     

(2+1)-dimensional ZK–Burgers equation with the generalized beta effect and its exact solitary solution

The (1 ＋1)-dimensional mathematical model had been extensively derived to describe Rossby solitary waves in a line in the past few decades. But as is well known, the (1 ＋1)-dimensional model cannot reflect the generation and evolution of Rossby solitary waves in a plane. In this paper, a (2 ＋1)-dimensional nonlinear Zakharov–Kuznetsov–Burgers equation is derived to describe the evolution of Rossby wave amplitude by using methods of multiple scales and perturbation expansions from the quasi-geostrophic potential vorticity equations with the generalized beta effect. The effects of the generalized beta and dissipation are presented by the Zakharov–Kuznetsov–Burgers equation. We also obtain the new solitary solution of the Zakharov–Kuznetsov equation when the dissipation is absent with the help of the Bernoulli equation, which is different from the common classical solitary solution. Based on the solution, the features of the variable coefficient are discussed by geometric figures Meanwhile, the approximate solitary solution of Zakharov–Kuznetsov–Burgers equation is given by using the homotopy perturbation method. And the amplitude of solitary waves changing with time is depicted by figures. Undoubtedly, these solitary solutions will extend previous results and better help to explain the feature of Rossby solitary waves.     

Convergence conditions for iterative methods seeking multi-component solitary waves with prescribed quadratic conserved quantities

We obtain linearized (i.e., non-global) convergence conditions for iterative methods that seek solitary waves with prescribed values of quadratic conserved quantities of multi-component Hamiltonian nonlinear wave equations. These conditions extend the ones found for single-component solitary waves in a recent publication by Yang and the present author. We also show that, and why, these convergence conditions coincide with dynamical stability conditions for ground-state solitary waves.Notably, our analysis applies regardless of whether the number of quadratic conserved quantities, s, equals or is less than the number of equations, S. To illustrate the situation when s < S, we use one of our iterative methods to find ground-state solitary waves in spin-1 Bose-Einstein condensates in a magnetic field (s = 2, S = 3).