几类非线性方程的对称、守恒律及精确解研究

发布时间：2018-04-02 17:53

本文选题：非线性发展方程　切入点：G　出处：《聊城大学》2017年硕士论文

【摘要】：本文主要运用李群理论,G'/G展开法,幂级数法等方法对几类非线性发展方程进行了研究,如变系数Riccati方程,广义的非线性耗散-色散方程,非线性Aceive耗散色散方程等,得到了这些方程一些新的精确解.第一章是通过齐次平衡原理和G'/G展开法对变系数Riccati方程进行研究,证明其存在G'/G解,通过进一步的求解得到部分满足特定条件的变系数Riccati方程的G'/G解,并例举一些满足条件的方程和对这些方程的求解.第二章应用李群方法对广义非线性Aceive耗散色散方程进行研究,得到该方程的李对称和群不变解,接下来对方程进行约化,再通过求解约化方程得到了该方程的一些新的精确解,例如幂级数解等,最后,利用向量场和伴随方程的方法得出广义非线性Aceive耗散色散方程的守恒律.第三章利用李群方法把1+1维的一般非线性耗散色散方程降维成常微分方程,在利用一些求解常微分方程的方法进行求解,进而求得一般非线性耗散色散方程一些新的精确解,其中包括G'/G解、幂级数解等,最后,利用向量场和伴随方程的方法得出非线性耗散色散方程的守恒律.由上可知,李群方法对于方程的降维具有很大的便利性和实用性,通过李群方法的反复的利用,最终可把高维的方程降低为一维的常微分方程,而对于常微分方程的求解是我们所熟悉的,我们可以通过各种方法来求解,如幂级数法、tanh法、辅助函数的方法等,这样就可以得到原高维方程的解.
[Abstract]:In this paper, several kinds of nonlinear evolution equations, such as variable coefficient Riccati equation, generalized nonlinear dissipative dispersion equation, nonlinear Aceive dissipative dispersion equation and so on, are studied by using Li Qun's theory and power series method.Some new exact solutions to these equations are obtained.In the first chapter, we study the Riccati equation with variable coefficients by using the homogeneous equilibrium principle and the G / G expansion method. We prove that there exists a G / G solution for the Riccati equation with variable coefficients, and obtain the G / G solution of the Riccati equation with variable coefficients partially satisfying the specific conditions.Some equations satisfying the conditions and their solutions are given.In the second chapter, the generalized nonlinear Aceive dissipative dispersion equation is studied by using Li Qun's method. The lie symmetry and group invariant solutions of the equation are obtained. Then the equation is reduced and some new exact solutions of the equation are obtained by solving the reduced equation.Finally, the conservation law of generalized nonlinear Aceive dissipative dispersion equation is obtained by the method of vector field and adjoint equation.In chapter 3, the general nonlinear dissipative dispersion equation of 11 dimension is reduced to an ordinary differential equation by Li Qun's method, and some new exact solutions of the general nonlinear dissipative dispersion equation are obtained by using some methods to solve the ordinary differential equation.It includes G / G solution, power series solution and so on. Finally, the conservation law of nonlinear dissipative dispersion equation is obtained by the method of vector field and adjoint equation.From the above, we can see that Li Qun's method has great convenience and practicability for reducing the dimension of the equation. By the repeated use of Li Qun's method, the high-dimensional equation can eventually be reduced to a one-dimensional ordinary differential equation.The solution of ordinary differential equation is familiar to us, we can solve it by various methods, such as power series method, auxiliary function method, etc. In this way, the solution of the original high-dimensional equation can be obtained.
【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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