几类非线性波动方程的可积性及孤立波解的谱稳定性研究

发布时间：2018-01-10 17:18

本文关键词：几类非线性波动方程的可积性及孤立波解的谱稳定性研究　出处：《昆明理工大学》2017年博士论文　论文类型：学位论文

【摘要】：非线性波广泛存在于自然界中,比如:水波、气体中的激波、等离子波、固体中的冲击波、星系中的密度波和地震波等。非线性波动方程是描述自然界中各种波动现象的重要数学物理模型,研究非线性波动方程的可积性、求解以及解的动力学行为,有助于人们揭示非线性波的传播规律,科学解释对应的自然现象,进一步推动非线性波理论的发展。本文研究在流体力学、等离子体物理和非线性光学中有重要应用的BKP方程、(2+1)维KdV方程、KP型系统和Sharma-Tasso-Olver方程的可积性、孤立波解及其动力学行为。主要研究内容和结果如下:1.基于双Bell多项式理论和Hirota双线性方法,研究BKP方程的双线性形式和孤立波解。通过引入微分约束条件和解耦技巧,得到了 BKP方程的几类Hirota双线性形式、Bell多项式型Backlund变换和Lax对;进而运用所得Hirota双线性形式得到了其多波解、Complexiton-解、亮-暗块状波解以及扭结-块状波相互作用解,并进一步研究其Complexiton-解和亮-暗块状波解之间的关系,发现亮-暗块状波解是Complexiton-解的极限,而Complexiton-解的解析式可由三角函数csc2(πx)的幂级数导出。此外,还通过Bell多项式型Lax对构造出BKP方程的守恒律。2.运用广义对称法,得到了 BKP方程的对称、KMV型李代数和守恒律。基于BKP方程的对称结构直接构造了 BKP方程的广义群不变解,运用广义群不变解,导出了 BKP方程的连续对称群和离散对称群。运用Painleve截断展开法,获得了 BKP方程的非自Backlund变换和非局部对称。3.基于Hirota双线性方法,研究实(2+1)维KdV方程和复KP型系统的块状波解。通过数值模拟研究发现,两类系统的块状波解都会产生时空偏转现象,而且在不同的参数条件下,块状波解会呈现出三类不同的时空结构。理论分析表明,平衡点分岔是导致块状波解时空偏转现象产生的原因之一。4.基于平面动力系统方法和Hirota双线性方法,研究了 STO方程扭结波解的存在性。运用能量估计方法,证明了其扭结波解是谱稳定的。通过拓展的同宿测试函数法得到了 STO方程的另一类扭结波相互作用解,数值模拟研究和理论分析表明,这类扭结波解聚变和裂变现象的产生并不依赖于色散系数α,而由图像平移参数(?)决定。α的符号决定着孤立波的传播方向:当α0时,孤立波向左传播;当α0时,孤立波向右传播。
[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.

【学位授予单位】：昆明理工大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O175.29

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