几类非线性发展方程不变解和守恒律的研究

发布时间：2018-04-08 15:54

本文选题：非线性发展方程　切入点：扩展的tanh函数展开法　出处：《聊城大学》2017年硕士论文

【摘要】：非线性发展方程(组)精确解的获得对物理、化学等多个领域解释复杂现象、解决难题具有重要的实际意义.它不但使问题可以进行定量研究,而且为定性理论分析等现实问题提供了必要的基础.因此其求解问题一直是该领域研究的重要课题之一.本文主要运用经典李群方法,结合扩展的tanh函数展开法和Riccati辅助方程方法以及幂级数展开法等研究了几类非线性发展方程,如一类KdV-mKdV方程、扩展的(2+1)维Jaulent-Miodek方程、扩展的Kadomtsov-Petviashvili-Benjamin-Bona-Mahoney(简写为KP-BBM)方程、(3+1)维Zakharov-Kuznetsov-Burgers(简写为ZKB)方程,并且得到了这些方程大量新的精确解和守恒律.第一章,利用齐次平衡法与(G￠/G)-展开法,求出一类KdV-mKdV方程不同情况下的显式解.结果表明,(G￠/G)-展开法和齐次平衡法的计算简单明了,对于其它某些非线性发展方程(组)同样适用.第二章,通过对直接对称法和计算机代数系统Maple的应用,求得了扩展的(2+1)维Jaulent-Miodek方程一些新的显式解,包括Weierstrass周期解、椭圆周期解、有理函数解等.最后,基于得到的对称和共轭方程,求得方程的守恒律。第三章,应用经典李群方法得到扩展的KP-Benjamin-Bona-Mahoney方程的李点对称和群不变解.结合辅助函数方法对约化方程进行求解,求得了KP-BBM方程一些新的精确解.并且求得了方程的守恒律.第四章,利用待定系数法计算(3+1)维Zakharov-Kuznetsov-Burgers方程的对称和约化方程.结合幂级数展开法和Riccati辅助方程方法等方法,得到方程一些新的显式解.在求得对称与共轭方程的基础上,给出了方程的守恒律。综上所述,本文的主要内容是把经典李群理论应用到非线性发展方程的求解过程中,利用经典李群方法求出方程的对称与相似约化方程.经过适当的变换,再借助不同的辅助方程方法对约化方程进行讨论,从而达到降维便于求解的目的。
[Abstract]:Nonlinear evolution equation (Group) exact solutions obtained from the physical, chemistry and other fields to explain the complex phenomenon, has important practical significance to solve the problem. It can not only make the problem of quantitative study, and provide a necessary basis for the practical problems of qualitative analysis. So the problem is always an important topic in the field of study. This paper mainly uses the classical Li Qun method and Riccati expansion method and the auxiliary equation method and power series of several kinds of nonlinear evolution equation method combined with the extended tanh function, such as a class of KdV-mKdV equation, the extended (2+1) - dimensional Jaulent-Miodek equation, the extended Kadomtsov-Petviashvili-Benjamin-Bona-Mahoney (KP-BBM) equation, (3+1 d) Zakharov-Kuznetsov-Burgers (ZKB) equation, and obtain abundant new exact solutions of these equations and conservation laws. In the first chapter, the use of Qi Time and balance method (G of /G) - expansion method for a class of KdV-mKdV equation explicit solutions under different conditions. The results showed that (G of /G) - expansion method and the homogeneous balance method is simple and clear, for some other nonlinear evolution equations (Group) applies. The second chapter, through the application of Maple method and computer algebra system directly symmetric, extended (2+1) obtained some new explicit solutions of Jaulent-Miodek equation, including Weierstrass elliptic periodic solutions, periodic solutions and rational solutions. Finally, the symmetries and adjoint equation based on the conservation law equation obtained. In the third chapter, by extended KP-Benjamin-Bona-Mahoney equation using the classical Li Qun method lie point symmetry and group invariant solutions. Combining the auxiliary function method was used to solve the equations, obtained some new exact solutions of KP-BBM equation are obtained. And the conservation law equation. In the fourth chapter, by the undetermined The calculation of (3+1) - dimensional Zakharov-Kuznetsov-Burgers equation and symmetry reduction equations. With the method of power series expansion and Riccati auxiliary equation method and other methods, obtained some new explicit solutions. Based on the obtained symmetry and adjoint equations on given conservation equation. To sum up, the main content of this paper is the classical Li Qun the theory is applied to solving nonlinear evolution equations, using the classical Li Qun method for equation of symmetry and similarity reduction equations. After the appropriate transformation, then using the auxiliary equation method of different equations are discussed, so as to achieve the purpose of dimension reduction is easy to solve.

【学位授予单位】：聊城大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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