流体力学和热传的非线性色散波方程的动力学及多孤立子解

发布时间：2018-03-31 00:29

本文选题：非线性波　切入点：Davey-Stewartson方程　出处：《华中师范大学》2017年博士论文

【摘要】：非线性色散水波是自然界中重要的可观察的现象之一。波浪通过材料介质(固体,液体或气体)波速传播,其方式和速度依赖于介质的弹性和惯性特性的。其研究还涉及流体动力学和对流热传递。本文的一部分研究在重力和表面张力效应下,平面水-空气界面上的表面重力波的传播。水波是波动区域中最引人入胜且变化最大的对象。数学和物理问题需要研究水波和他们在海滩上的破裂现象。本论文另一部分重点是研究在重力效应和垂直温度梯度变化的作用下,接触空气的水平流体层中的表面波的传播。通过最低阶扰动化归技术方法,非线性PDE类可以归结到更容易处理的单个非线性方程。研究了在表面张力和重力作用下,有限深度的流体的浅水(SW)模型的三维非线性色散波,并导出了2-D谐波满足的Davey-Stewartson(DS)方程。通过对该模型的线性部分的分析得到了方程的色散性质。我们也对DS方程的守恒定律进行了详细的推导和讨论。应用了Painleve分析,我们不仅研究DS方程的可积性,而且通过截断的Painleve展开来构建Backlund变换。最后,通过采用Backklund变换,哈密尔顿算法和改进的(G'/G)级数展开方法研究了DS方程,并获得了新的行波孤立和扭结波解。利用最简单的方程方法,我们得到了精确的行波解和一个广义DS模型多孤立子形式的解。该结果表明,随着Ursell参数增加得越大波幅就减小的越多。同时波剖面与时间有相似的趋势。它还揭示了结果与势能守恒的一致性随着Ursell参数的增加而增加。在哈密尔顿算法中,我们发现波的振幅随着能量常数的增加而增加。进一步地,为了揭示其稳定性,相平面法被应用来分析DS模型推导的非线性一阶方程。我们研究了在重力场和垂直温度梯度效应下,接触空气的水平流体层的表面波问题。我们提出了描述问题的控制方程并将其转换为非线性发展方程,该方程是扰动的Korteweg-de Vries(pKdV)方程。研究了在对流流体环境中该方程的长程表面波的演化,构建和讨论了pKdV方程的色散关系及其概念。应用Painleve分析来检验pKdV方程的可积性,并建立该方程的Backlund变换形式。使用Backlund变换,Bernoulli,Riccati最简单的方程方法,Burgers方程和新形式的因式分解等方法,我们发现了新的行波解和pKdV方程的多个孤子解的一般形式。论文的最后一部分涉及研究耦合型立方-五次的复Ginzburg-Landau(cc-qcGL)方程。这些方程可用于描述对流性二相流体在周期性空间-时间模式下的缓慢折叠性非线性演化。我们首先构建了模型的色散关系及其性质。通过Painleve分析不仅用于检验了模型的可积性,而且还用于建立Backlund转换形式。此外,通过在后两种模型中使用的Back-lund变换和最简单的方程方法,获得了新的行波解和cc-qcGL方程的多孤立子解的一般形式。通过使用各种分析方法研究了所有模型的解,并在几个3-D和2-D图形中进行了说明,显示了流动中的冲击和孤立波性质。
[Abstract]:Nonlinear dispersive wave is one of the important observation of the nature of the phenomenon. The wave through the material medium (solid, liquid or gas) wave propagation, elastic and inertial characteristics and its speed depends on the media. The study also relates to fluid dynamics and convective heat transfer. A part of the research on gravity and surface tension under the effect of surface water air interface on the surface gravity wave propagation. It is the largest and most fascinating object changes in mathematics and physics. The fluctuation of regional problems need to be studied and their waves on the beach rupture. The second part is focus on research in the gravity effect and the vertical temperature gradient changes under the action of surface wave propagation, a horizontal fluid layer in contact with the air. Through to the lowest order perturbation technique, nonlinear PDE can be attributed to a single nonlinear processing more easily In the research process. The effects of surface tension and gravity, the shallow fluid of finite depth (SW) three-dimensional nonlinear dispersive wave model, and deduced the 2-D harmonic content Davey-Stewartson (DS) equation. Based on the analysis of the linear part of the model of the dispersion equation. We also get the quality of DS equation the conservation laws are derived and discussed in detail. The application of the Painleve analysis, we not only study the integrability of DS equation, and the truncated Painleve expansion to construct Backlund transform. Finally, by using the Backklund transform, Hamilton algorithm and improved (G'/G) series expansion method of the DS equation, and obtain the traveling wave isolated and kink wave solutions. Using the new equation of the most simple method, we obtain the exact traveling wave solutions and a generalized DS model of multi soliton solution. The results show that with the increase of Ursell parameters The large amplitude decreases more. At the same time the wave profile and time have a similar trend. It also reveals the consistency of results and potential energy conservation increases with the increase of Ursell parameters in Hamilton. In the algorithm, we found that the amplitude of the wave increases with increasing energy constant. Furthermore, in order to reveal the stability of phase plane by analysis of the nonlinear DS model is derived as a first-order equation. We studied in the gravity field and the vertical temperature gradient effect, surface level fluid layer exposed to air. We propose a control equation describing the problem and transform it into a nonlinear evolution equation, the equation is perturbed Korteweg-de Vries (pKdV) study on the evolution equation. The equation of the long-range surface wave in the fluid convection in the environment, construction and discussed the dispersion relation and the concept of pKdV equation. The application of Painleve analysis to test pKdV The integrability of the equation, and establish the Backlund transform of the equation. Using the Backlund transform, Bernoulli equation, the easiest way to Riccati, the Burgers equation and the new form of the factorization method, we find that the general form of a number of new traveling wave solutions and soliton solutions of the pKdV equation. The last part of the thesis relates to study on coupling of cubic - five complex Ginzburg-Landau (cc-qcGL) equation. These equations can be used to describe the temporal patterns in the periodic space - convective two-phase flow under the slow folding of nonlinear evolution. We constructed a model of the dispersion relations and properties. Through the analysis of the Painleve is not only used to test the model integrable. But also for the establishment of Backlund forms. In addition, the equation method of Back-lund transform in use after the two model and the most simple, get a new travelling wave solutions of cc-qcGL equation The general form of multiple soliton solutions is studied. The solutions of all the models are studied by using various analytical methods. It is illustrated in several 3-D and 2-D graphs, showing the characteristics of shock and solitary waves in flow.

【学位授予单位】：华中师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175

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