【摘要】：淺水波系統(tǒng)的研究不僅是應(yīng)用數(shù)學(xué)和數(shù)學(xué)物理的一個(gè)重要組成部分,而且是非線性偏微分方程研究中的一個(gè)重要課題.本文以廣義兩分量Dullin-Gottwald-Holm(GDGH2)淺水波系統(tǒng)為模型,研究了該系統(tǒng)及其推廣形式的一類自相似解,通過構(gòu)造相應(yīng)的Emden方程,分析了解的全局存在性及有限時(shí)間的爆破現(xiàn)象.在數(shù)學(xué)物理研究中,精確解的構(gòu)造不僅能讓我們深入的了解方程本身的性質(zhì),還可以有效幫助刻畫一些非線性現(xiàn)象.雖然構(gòu)造方法并不唯一,但其主旨均是簡化非線性問題.常用的求解方法有Darboux變換,反散射方法,Backlund變換,Hirota雙線性方法,Painleve奇性分析法,分離變量法等.本文主要利用分離變量法,結(jié)合擾動(dòng)方法和特征線方法研究了 GDGH2系統(tǒng)及推廣的GDGH2系統(tǒng)自相似解的結(jié)構(gòu)與性質(zhì).主要研究內(nèi)容如下:首先,利用分離變量法,構(gòu)造了 GDGH2系統(tǒng)中具有橢圓對(duì)稱及drift結(jié)構(gòu)的自相似解,并分析了解的全局存在性及爆破現(xiàn)象.其次,對(duì)GDGH2系統(tǒng)進(jìn)行推廣,構(gòu)造了推廣的GDGH2系統(tǒng)的自相似爆破解,并分析了解的特性.進(jìn)一步,利用擾動(dòng)方法和特征線法,構(gòu)造了兩類精確解的結(jié)構(gòu),論述了解的性質(zhì)及研究意義.
[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.
【學(xué)位授予單位】：西北大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.29

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相關(guān)期刊論文 前2條

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本文編號(hào)：2217727