MKdV族与MNW族的新解及KdV型方程解的新算法

发布时间：2018-07-16 21:39

【摘要】：从数学角度来看,在非线性偏微分方程中,孤子是一类特殊的解.伴随着孤立子理论的发展,寻找非线性偏微分方程的孤波解是孤子理论中的一个有意义的工作,具有重要的理论与实际应用价值,也是一个非常值得探究的课题.Hirota双线性方法、F展开法和指数函数法是非线性偏微分方程的三个构造性求解方法,这样的方法不依赖于方程的谱问题或Lax对,使得求解过程更加直接.本文在介绍孤子理论的研究历史及其发展状况的基础上,一方面将Hirota双线性方法和F展开法分别推广应用于新的mKdV族和MNW族,另一方面利用所提出的指数函数法的新算法求解一个KdV型方程.本文的主要工作如下:第二章利用Hirota双线性方法求解一个以常系数mKdV族为特例的新的变系数mKdV族.首先,通过引入有效变换将此变系数mKdV族进行双线性化,得到其双线性形式.其次基于所得的双线性形式,利用截断展开技术构造出变系数mKdV族的新单孤子解、双孤子解、三孤子解,同时归纳出N孤子解的一般数学表达式,并对所得部分孤子解的演化行为特征进行图像模拟。第三章基于F展开法构造非线性偏微分方程精确解的具体步骤,将F展开法推广应用于一个新的MNW方程族,获得许多新的Jacobi椭圆函数解.在模趋于1和0的极限情形下,由这些所获得的Jacobi椭圆函数解得到许多新的双曲函数解和三角函数解.为便于讨论所得精确解的局域空间结构以及解的动力演化,本部分还插入了一些解的二维和三维图像。第四章首先概述了本文所提出指数函数法的一个直接算法,此算法的优点在于与原指数函数法相比较在计算过程中出现“中间表达式膨胀”的程度较小.其次,为进一步找到此算法的新的应用实例,考虑了一个变系数KdV型方程,结果获得此KdV型方程含多个系数函数形式的精确孤波解,从中显示出所提出的指数函数法的一个新算法的优越性及其广泛的应用范围。
[Abstract]:From the mathematical point of view, soliton is a special kind of solution in nonlinear partial differential equation. With the development of soliton theory, finding soliton solutions of nonlinear partial differential equations is a meaningful work in soliton theory, which has important theoretical and practical application value. Hirota bilinear method / F expansion method and exponential function method are three constructive methods for solving nonlinear partial differential equations, which do not depend on the spectral problems or lax pairs of equations. The solution process is more direct. On the basis of introducing the history and development of soliton theory, on the one hand, the Hirota bilinear method and the F-expansion method are extended to the new mKdV and MNW families, respectively. On the other hand, a new algorithm of exponential function method is proposed to solve a KDV type equation. The main work of this paper is as follows: in chapter 2, we use Hirota bilinear method to solve a new variable coefficient mKdV family with constant coefficient mKdV as a special case. Firstly, the mKdV family of variable coefficients is bilinear by introducing effective transformation. Secondly, based on the bilinear form obtained, the new single-soliton solution, double-soliton solution and three-soliton solution of mKdV family with variable coefficients are constructed by using truncation expansion technique, and the general mathematical expressions of N-soliton solution are also summarized. The evolutionary behavior of some soliton solutions is simulated by image simulation. In chapter 3, based on the concrete steps of constructing exact solutions of nonlinear partial differential equations by F-expansion method, the F-expansion method is extended to a new family of MNW equations and many new solutions of Jacobi elliptic functions are obtained. In the case of a module approaching to the limit of 1 and 0, many new solutions of hyperbolic and trigonometric functions are obtained from the Jacobi elliptic function solutions obtained from these solutions. To facilitate the discussion of the local spatial structure of the exact solution and the dynamic evolution of the solution, some 2D and 3D images of the solution are also inserted in this part. In the fourth chapter, a direct algorithm of exponential function method proposed in this paper is introduced. The advantage of this algorithm is that compared with the original exponential function method, the degree of "intermediate expression expansion" in the calculation process is smaller. Secondly, in order to find a new application example of this algorithm, a variable coefficient KDV type equation is considered. The exact solitary wave solution of the KDV equation with multiple coefficient functions is obtained. It shows the superiority of a new algorithm of exponential function method and its wide range of application.
【学位授予单位】：渤海大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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