本文關(guān)鍵詞：幾類非線性波動方程的可積性及孤立波解的譜穩(wěn)定性研究　出處：《昆明理工大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 非線性波動方程 Hirota雙線性形式 對稱分析 扭結(jié)孤立波 塊狀波 譜穩(wěn)定性

【摘要】：非線性波廣泛存在于自然界中,比如:水波、氣體中的激波、等離子波、固體中的沖擊波、星系中的密度波和地震波等。非線性波動方程是描述自然界中各種波動現(xiàn)象的重要數(shù)學(xué)物理模型,研究非線性波動方程的可積性、求解以及解的動力學(xué)行為,有助于人們揭示非線性波的傳播規(guī)律,科學(xué)解釋對應(yīng)的自然現(xiàn)象,進(jìn)一步推動非線性波理論的發(fā)展。本文研究在流體力學(xué)、等離子體物理和非線性光學(xué)中有重要應(yīng)用的BKP方程、(2+1)維KdV方程、KP型系統(tǒng)和Sharma-Tasso-Olver方程的可積性、孤立波解及其動力學(xué)行為。主要研究內(nèi)容和結(jié)果如下:1.基于雙Bell多項式理論和Hirota雙線性方法,研究BKP方程的雙線性形式和孤立波解。通過引入微分約束條件和解耦技巧,得到了 BKP方程的幾類Hirota雙線性形式、Bell多項式型Backlund變換和Lax對;進(jìn)而運用所得Hirota雙線性形式得到了其多波解、Complexiton-解、亮-暗塊狀波解以及扭結(jié)-塊狀波相互作用解,并進(jìn)一步研究其Complexiton-解和亮-暗塊狀波解之間的關(guān)系,發(fā)現(xiàn)亮-暗塊狀波解是Complexiton-解的極限,而Complexiton-解的解析式可由三角函數(shù)csc2(πx)的冪級數(shù)導(dǎo)出。此外,還通過Bell多項式型Lax對構(gòu)造出BKP方程的守恒律。2.運用廣義對稱法,得到了 BKP方程的對稱、KMV型李代數(shù)和守恒律。基于BKP方程的對稱結(jié)構(gòu)直接構(gòu)造了 BKP方程的廣義群不變解,運用廣義群不變解,導(dǎo)出了 BKP方程的連續(xù)對稱群和離散對稱群。運用Painleve截斷展開法,獲得了 BKP方程的非自Backlund變換和非局部對稱。3.基于Hirota雙線性方法,研究實(2+1)維KdV方程和復(fù)KP型系統(tǒng)的塊狀波解。通過數(shù)值模擬研究發(fā)現(xiàn),兩類系統(tǒng)的塊狀波解都會產(chǎn)生時空偏轉(zhuǎn)現(xiàn)象,而且在不同的參數(shù)條件下,塊狀波解會呈現(xiàn)出三類不同的時空結(jié)構(gòu)。理論分析表明,平衡點分岔是導(dǎo)致塊狀波解時空偏轉(zhuǎn)現(xiàn)象產(chǎn)生的原因之一。4.基于平面動力系統(tǒng)方法和Hirota雙線性方法,研究了 STO方程扭結(jié)波解的存在性。運用能量估計方法,證明了其扭結(jié)波解是譜穩(wěn)定的。通過拓展的同宿測試函數(shù)法得到了 STO方程的另一類扭結(jié)波相互作用解,數(shù)值模擬研究和理論分析表明,這類扭結(jié)波解聚變和裂變現(xiàn)象的產(chǎn)生并不依賴于色散系數(shù)α,而由圖像平移參數(shù)(?)決定。α的符號決定著孤立波的傳播方向:當(dāng)α0時,孤立波向左傳播;當(dāng)α0時,孤立波向右傳播。
[Abstract]:Nonlinear waves exist widely in nature, such as wave, shock wave, gas plasma wave, shock wave in solid, Galaxy density wave and seismic wave. The nonlinear wave equation is an important mathematical model describing various wave phenomena in nature, the research of nonlinear wave equation integrability, solutions and solutions the dynamic behavior, help people to reveal the propagation of nonlinear waves, corresponding scientific explanations of natural phenomena, to further promote the development of nonlinear wave theory. This paper studies on fluid mechanics, BKP equation have important applications in plasma physics and nonlinear optics, (2+1) - dimensional KdV equation, KP system and the Sharma-Tasso-Olver equation can be integrability, solitary wave solutions and its dynamic behavior. The main research contents and results are as follows: 1. based on double Bell polynomial theory and Hirota bilinear method, BKP bilinear equation research Form and solitary wave solutions. By introducing differential constraints and decoupling technique, we obtain several classes of Hirota bilinear form of BKP equation, Bell polynomial Backlund transform and Lax; and then the multi solution, Complexiton- solution is obtained by using the Hirota bilinear form, bright dark blocks and kink wave solutions - bulk wave interaction the solution, and further study of the Complexiton- solution and the bright dark relationship between massive wave solutions, found that the bright dark wave solution is Complexiton- solution bulk limit, and analytical solutions of Complexiton- type by trigonometric function csc2 (n x) power series is derived. In addition, through the Bell Lax BKP polynomial equation.2. conservation law using the generalized symmetry method of structure, the symmetry of the BKP equation, the KMV type lie algebra and conservation laws. The symmetrical structure based on BKP equation directly constructed BKP equations and generalized group invariant solutions, using generalized invariant solution, Are continuous symmetry group BKP equation and discrete symmetry group. Using Painleve truncated expansion method, obtained the BKP equation of non Hirota bilinear method based on Backlund transform and non locally symmetric.3. (2+1) on massive wave solutions of the two-dimensional KdV equation and the complex KP system. Simulation results showed that the massive wave two types of systems will produce space-time deflection phenomenon, and under different parametric conditions, massive wave solutions will exhibit three different space-time structure. The theoretical analysis shows that the equilibrium bifurcation is one of the reasons resulting in massive.4. wave solutions of the space-time deflection phenomenon of planar dynamical systems method and the Hirota bilinear method based on research the STO equation of kink wave solution. By using the energy estimation method, proved the kink wave solutions is spectrum stability. Through the homoclinic test function method has been extended STO equation of another type of kink wave The interaction of solution, numerical simulation and theoretical analysis show that the kink waves of fusion and fission phenomenon is not dependent on the dispersion coefficient, and by the image translation parameters (?). Alpha symbol determines the direction of propagation of solitary waves: when alpha 0, solitary wave spread to the left; when alpha 0 when the solitary wave spread to the right.

【學(xué)位授予單位】：昆明理工大學(xué)
【學(xué)位級別】：博士
【學(xué)位授予年份】：2017
【分類號】：O175.29

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