本文選題：非線性波　切入點(diǎn)：Davey-Stewartson方程　出處：《華中師范大學(xué)》2017年博士論文

【摘要】：非線性色散水波是自然界中重要的可觀察的現(xiàn)象之一。波浪通過材料介質(zhì)(固體,液體或氣體)波速傳播,其方式和速度依賴于介質(zhì)的彈性和慣性特性的。其研究還涉及流體動(dòng)力學(xué)和對(duì)流熱傳遞。本文的一部分研究在重力和表面張力效應(yīng)下,平面水-空氣界面上的表面重力波的傳播。水波是波動(dòng)區(qū)域中最引人入勝且變化最大的對(duì)象。數(shù)學(xué)和物理問題需要研究水波和他們?cè)诤┥系钠屏熏F(xiàn)象。本論文另一部分重點(diǎn)是研究在重力效應(yīng)和垂直溫度梯度變化的作用下,接觸空氣的水平流體層中的表面波的傳播。通過最低階擾動(dòng)化歸技術(shù)方法,非線性PDE類可以歸結(jié)到更容易處理的單個(gè)非線性方程。研究了在表面張力和重力作用下,有限深度的流體的淺水(SW)模型的三維非線性色散波,并導(dǎo)出了2-D諧波滿足的Davey-Stewartson(DS)方程。通過對(duì)該模型的線性部分的分析得到了方程的色散性質(zhì)。我們也對(duì)DS方程的守恒定律進(jìn)行了詳細(xì)的推導(dǎo)和討論。應(yīng)用了Painleve分析,我們不僅研究DS方程的可積性,而且通過截?cái)嗟腜ainleve展開來構(gòu)建Backlund變換。最后,通過采用Backklund變換,哈密爾頓算法和改進(jìn)的(G'/G)級(jí)數(shù)展開方法研究了DS方程,并獲得了新的行波孤立和扭結(jié)波解。利用最簡單的方程方法,我們得到了精確的行波解和一個(gè)廣義DS模型多孤立子形式的解。該結(jié)果表明,隨著Ursell參數(shù)增加得越大波幅就減小的越多。同時(shí)波剖面與時(shí)間有相似的趨勢(shì)。它還揭示了結(jié)果與勢(shì)能守恒的一致性隨著Ursell參數(shù)的增加而增加。在哈密爾頓算法中,我們發(fā)現(xiàn)波的振幅隨著能量常數(shù)的增加而增加。進(jìn)一步地,為了揭示其穩(wěn)定性,相平面法被應(yīng)用來分析DS模型推導(dǎo)的非線性一階方程。我們研究了在重力場(chǎng)和垂直溫度梯度效應(yīng)下,接觸空氣的水平流體層的表面波問題。我們提出了描述問題的控制方程并將其轉(zhuǎn)換為非線性發(fā)展方程,該方程是擾動(dòng)的Korteweg-de Vries(pKdV)方程。研究了在對(duì)流流體環(huán)境中該方程的長程表面波的演化,構(gòu)建和討論了pKdV方程的色散關(guān)系及其概念。應(yīng)用Painleve分析來檢驗(yàn)pKdV方程的可積性,并建立該方程的Backlund變換形式。使用Backlund變換,Bernoulli,Riccati最簡單的方程方法,Burgers方程和新形式的因式分解等方法,我們發(fā)現(xiàn)了新的行波解和pKdV方程的多個(gè)孤子解的一般形式。論文的最后一部分涉及研究耦合型立方-五次的復(fù)Ginzburg-Landau(cc-qcGL)方程。這些方程可用于描述對(duì)流性二相流體在周期性空間-時(shí)間模式下的緩慢折疊性非線性演化。我們首先構(gòu)建了模型的色散關(guān)系及其性質(zhì)。通過Painleve分析不僅用于檢驗(yàn)了模型的可積性,而且還用于建立Backlund轉(zhuǎn)換形式。此外,通過在后兩種模型中使用的Back-lund變換和最簡單的方程方法,獲得了新的行波解和cc-qcGL方程的多孤立子解的一般形式。通過使用各種分析方法研究了所有模型的解,并在幾個(gè)3-D和2-D圖形中進(jìn)行了說明,顯示了流動(dòng)中的沖擊和孤立波性質(zhì)。
[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.

【學(xué)位授予單位】：華中師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

【參考文獻(xiàn)】

相關(guān)期刊論文 前3條

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本文編號(hào)：1688358