多分量浅水波系统解的结构
发布时间：2018-09-01 16:23
【摘要】：浅水波系统的研究不仅是应用数学和数学物理的一个重要组成部分,而且是非线性偏微分方程研究中的一个重要课题.本文以广义两分量Dullin-Gottwald-Holm(GDGH2)浅水波系统为模型,研究了该系统及其推广形式的一类自相似解,通过构造相应的Emden方程,分析了解的全局存在性及有限时间的爆破现象.在数学物理研究中,精确解的构造不仅能让我们深入的了解方程本身的性质,还可以有效帮助刻画一些非线性现象.虽然构造方法并不唯一,但其主旨均是简化非线性问题.常用的求解方法有Darboux变换,反散射方法,Backlund变换,Hirota双线性方法,Painleve奇性分析法,分离变量法等.本文主要利用分离变量法,结合扰动方法和特征线方法研究了 GDGH2系统及推广的GDGH2系统自相似解的结构与性质.主要研究内容如下:首先,利用分离变量法,构造了 GDGH2系统中具有椭圆对称及drift结构的自相似解,并分析了解的全局存在性及爆破现象.其次,对GDGH2系统进行推广,构造了推广的GDGH2系统的自相似爆破解,并分析了解的特性.进一步,利用扰动方法和特征线法,构造了两类精确解的结构,论述了解的性质及研究意义.
[Abstract]:The study of shallow water wave systems is not only an important part of applied mathematics and mathematical physics, but also an important subject in the study of nonlinear partial differential equations. In this paper, the generalized two-component Dullin-Gottwald-Holm (GDGH2) shallow water wave system is used as a model, and a class of self-similar solutions of the system and its extended form are studied. By constructing the corresponding Emden equation, the global existence of the solution and the phenomenon of finite time blasting are analyzed. In the study of mathematical physics, the construction of exact solutions can not only help us to understand the properties of the equation itself, but also help to describe some nonlinear phenomena. Although the construction method is not unique, its main purpose is to simplify the nonlinear problem. The commonly used methods are Darboux transform, Backlund transform and Hirota bilinear method, Painleve singularity analysis method, the method of separating variables and so on. In this paper, the structure and properties of self-similar solutions of GDGH2 system and extended GDGH2 system are studied by using the method of separating variables and the method of perturbation and characteristic line. The main research contents are as follows: firstly, the self-similar solutions with ellipse symmetry and drift structure in GDGH2 systems are constructed by using the method of separating variables, and the global existence and blasting phenomena of the solutions are analyzed. Secondly, we generalize the GDGH2 system and construct the self-similar explosive crack of the extended GDGH2 system, and analyze the characteristics of the solution. Furthermore, by using perturbation method and characteristic line method, the structure of two kinds of exact solutions is constructed, and the properties and significance of the solution are discussed.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
本文编号：2217727
[Abstract]:The study of shallow water wave systems is not only an important part of applied mathematics and mathematical physics, but also an important subject in the study of nonlinear partial differential equations. In this paper, the generalized two-component Dullin-Gottwald-Holm (GDGH2) shallow water wave system is used as a model, and a class of self-similar solutions of the system and its extended form are studied. By constructing the corresponding Emden equation, the global existence of the solution and the phenomenon of finite time blasting are analyzed. In the study of mathematical physics, the construction of exact solutions can not only help us to understand the properties of the equation itself, but also help to describe some nonlinear phenomena. Although the construction method is not unique, its main purpose is to simplify the nonlinear problem. The commonly used methods are Darboux transform, Backlund transform and Hirota bilinear method, Painleve singularity analysis method, the method of separating variables and so on. In this paper, the structure and properties of self-similar solutions of GDGH2 system and extended GDGH2 system are studied by using the method of separating variables and the method of perturbation and characteristic line. The main research contents are as follows: firstly, the self-similar solutions with ellipse symmetry and drift structure in GDGH2 systems are constructed by using the method of separating variables, and the global existence and blasting phenomena of the solutions are analyzed. Secondly, we generalize the GDGH2 system and construct the self-similar explosive crack of the extended GDGH2 system, and analyze the characteristics of the solution. Furthermore, by using perturbation method and characteristic line method, the structure of two kinds of exact solutions is constructed, and the properties and significance of the solution are discussed.
【学位授予单位】：西北大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
【参考文献】
相关期刊论文 前2条
1 王昊;;Dullin-Gottwald-Holm浅水波系统中具椭圆对称的自相似解[J];西北大学学报(自然科学版);2016年05期
2 闫璐;时振华;王昊;康静;;INVARIANT SUBSPACES AND GENERALIZED FUNCTIONAL SEPARABLE SOLUTIONS TO THE TWO-COMPONENT b-FAMILY SYSTEM[J];Acta Mathematica Scientia(English Series);2016年03期
，本文编号：2217727
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