修正的简单方程法的应用以及变系数李方程组的研究
本文选题：修正的简单方程法 + 精确解 ； 参考：《内蒙古师范大学》2017年硕士论文
【摘要】：众多学科领域中出现的大量非线性现象一般都可以用非线性发展方程来刻画,因而用非线性模型来反映客观世界成为非线性科学研究的一股主流。于是寻找非线性发展方程精确解的问题自然成为直观深刻地去通晓非线性模型的物理意义及其性质所不可或缺的途径之一。正因为如此,建立寻找非线性发展方程精确解的新方法,借助或改进原有方法而给出非线性发展方程的新的精确解等问题能够在理论上帮助人们理解实际物理问题所蕴含的性质和特性,在应用方面能够提供新的方法和不同技巧等意义.此外,非线性模型的可积性问题是孤立子理论研究的另一重要课题,与此联系的可积系统的Lax对是将非线性方程的求解问题转化为线性方程的求解问题的桥梁,而Backlund变换则是构造非线性方程精确解的有效工具。因此,对于具体的非线性方程给出它的Lax对与Backlund变换不仅为可积系统各种性质的研究奠定理论基础,同时也对构造非线性方程精确解提供工具的作用.本文工作将围绕上述课题主要研究非线性发展方程(组)的求解问题并给出sine-Gordon方程、广义的变系数KdV-mKdV方程以及(2+1)维色散长波方程组的精确解,同时考虑变系数李方程组的Painleve性质、Lax对、Backlund变换与精确解的构造等问题。本文具体内容安排如下第一章为绪论,分别对Painleve分析、变系数非线性发展方程的研究现状、修正的简单方程法、辅助方程法作一简单介绍,并简短介绍本文的主要研究工作.第二章将借助修正的简单方程法给出sine-Gordon方程、广义的变系数KdV-mKdV方程以及(2+1)维色散长波方程组的精确解,得到其对应的精确孤波解.第三章研究变系数李方程组,证明变系数李方程组具有Painleve性质,给出它的Lax对、自Backlund变换以及精确解.第四章将分别借助Raccati辅助方程法和扩展的G'/G展开法给出(2+1)维AKNS方程以及高阶色散NLS方程的精确解,包括周期解、孤波解以及有理解.第五章,对全文的具体工作进行总结,并对今后开展的工作提出了详细的计划.
[Abstract]:A large number of nonlinear phenomena in many disciplines can be described by nonlinear evolution equations, so the nonlinear model to reflect the objective world has become the mainstream of nonlinear science research. Therefore, finding the exact solution of nonlinear evolution equation is naturally one of the indispensable ways to understand the physical meaning and properties of nonlinear model directly and profoundly. Because of this, a new method for finding exact solutions of nonlinear evolution equations is established. The new exact solutions of nonlinear evolution equations can help people understand the properties and characteristics of practical physical problems in theory by using or improving the original methods. It can provide new methods and different skills in application. In addition, the integrability problem of nonlinear model is another important subject of soliton theory. The Lax pair of integrable system connected with this problem is a bridge to transform the solving problem of nonlinear equation into the solving problem of linear equation. Backlund transformation is an effective tool for constructing exact solutions of nonlinear equations. Therefore, for specific nonlinear equations, the Lax pair and Backlund transformation not only lay a theoretical foundation for the study of various properties of integrable systems, but also provide a tool for constructing exact solutions of nonlinear equations. In this paper, the solution of nonlinear evolution equations (systems) is studied and the exact solutions of the sine-Gordon equation, the generalized KdV-mKdV equation with variable coefficients and the long-wave equations of dispersion in the first dimension are given. At the same time, we consider the Painleve property of lie equations with variable coefficients and the construction of exact solutions and so on. The main contents of this paper are as follows: the first chapter is the introduction, which gives a brief introduction to Painleve analysis, variable coefficient nonlinear evolution equation, modified simple equation method and auxiliary equation method, and briefly introduces the main research work of this paper. In the second chapter, the exact solutions of the sine-Gordon equation, the generalized KdV-mKdV equation with variable coefficients and the long-wave equations with dispersion of 21) dimension are given by using the modified simple equation method, and the corresponding exact solitary wave solutions are obtained. In chapter 3, we study the lie equations with variable coefficients. We prove that the lie equations with variable coefficients have Painleve properties, and give their Lax pairs, self- transformations and exact solutions. In chapter 4, the exact solutions of AKNS equation and higher order dispersive NLS equation, including periodic solution, solitary wave solution and understanding, are obtained by using the Raccati auxiliary equation method and the extended G / G expansion method, respectively. Chapter five summarizes the specific work of the paper and puts forward a detailed plan for the future work.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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