基于双线性和动力系统方法的非线性发展方程的解析研究

发布时间：2019-02-15 23:34

【摘要】：非线性科学是研究非线性问题共性的一门新的交叉学科,是自然科学领域中的一门学科。我们所研究的非线性发展方程主要是源于流体力学、等离子体物理、非线性光学、凝聚态物理、生物学等领域。如今,许多方法被应用于各类非线性发展方程的求解中,例如反散射法、Backlund法、Darboux变换法,Hirota双线性法、Painleve分析法、几何分支理论以及同宿呼吸子极限法等等。本文正是以非线性发展方程的理论为基础,运用Hirota双线性法、Painleve分析法、几何分支理论和同宿呼吸子极限法,借助计算机符号计算研究几个非线性发展方程,获得其解析解,并进一步研究其解的基本性质。本文的内容安排如下：(1)运用几何分支理论和动力系统定性理论研究了一个带克尔项的非线性Schrodinger方程,该方程描述光波在非线性光纤中的传输过程。通过行波变换,我们将偏微分方程转化为常微分方程,画出轨线图,判别解的类型,得到孤波解和椭圆函数周期解。(2)其次,研究了耦合修正Korteweg-de Vries方程组,该方程组描述了水波在传输过程中的相互作用。处理耦合修正Korteweg-de Vries方程组,通过两次行波变换,化为常微分系统,并进行定性理论分析。在不同的参数下,我们得到六组轨线图,根据轨线图的性质,得到孤波解和椭圆函数解。此外,运用扩展的椭圆函数法,得到新的椭圆函数解。(3)最后本文研究了一种特殊的水波——畸形波。研究了(2+1)维Kadomtsev-Petviashvili方程和(1+1)维对称正则长波方程,在Hirota双线性方法的基础上,借助符号计算,运用同宿呼吸子极限法,求得方程呼吸子解,进一步分析得到方程的畸形波解。综上所述,本文主要结合解析方法和计算机符号计算,跨学科地分析了在光通信和流体力学等领域中具有重要研究价值的一些非线性发展方程,包括对它们的解的性质的研究。本文中用到的研究方法,也可以应用到其它耦合及高阶非线性模型的研究上。另外,对于文中得到的研究结果,希望能为相关领域的研究提供一些理论上的帮助。
[Abstract]:Nonlinear science is a new interdiscipline to study the commonness of nonlinear problems, and it is also a subject in the field of natural science. The nonlinear evolution equations are mainly derived from hydrodynamics, plasma physics, nonlinear optics, condensed matter physics, biology and so on. Nowadays, many methods are applied to solve nonlinear evolution equations, such as backscattering method, Backlund method, Darboux transformation method, Hirota bilinear method, Painleve analysis method, geometric bifurcation theory, homoclinic respiratory limit method and so on. In this paper, based on the theory of nonlinear evolution equation, using Hirota bilinear method, Painleve analysis method, geometric bifurcation theory and homoclinic respiratory limit method, several nonlinear evolution equations are studied by means of computer symbolic calculation, and their analytical solutions are obtained. The basic properties of the solution are further studied. The contents of this paper are as follows: (1) A nonlinear Schrodinger equation with Kerr term is studied by using the geometric bifurcation theory and the qualitative theory of dynamical systems. The equation describes the propagation process of light waves in nonlinear optical fibers. Through traveling wave transformation, we transform partial differential equation into ordinary differential equation, draw derailment diagram, distinguish the type of solution, obtain solitary wave solution and periodic solution of elliptic function. (2) secondly, we study the coupled modified Korteweg-de Vries equations. The equations describe the interaction of water waves in the process of transmission. This paper deals with coupled modified Korteweg-de Vries equations and transforms them into ordinary differential systems by twice traveling wave transformation, and makes qualitative theoretical analysis. Under different parameters, we obtain six sets of orbital diagrams. According to the properties of the orbital diagrams, we obtain solitary wave solutions and elliptic function solutions. In addition, a new elliptic function solution is obtained by using the extended elliptic function method. (3) in the end, a special water wave, deformity wave, is studied. In this paper, the (21) dimensional Kadomtsev-Petviashvili equation and (11) dimensional symmetric regular long wave equation are studied. On the basis of Hirota bilinear method, by means of symbolic calculation and homoclinic respiratory limit method, the authors obtain the solutions of the equation respiratory elements. The deformable wave solution of the equation is obtained by further analysis. To sum up, some nonlinear evolution equations, including the properties of their solutions, which are of great value in the field of optical communication and fluid mechanics, are mainly analyzed in this paper by means of analytical methods and computer symbolic calculations. The methods used in this paper can also be applied to the study of other coupling and high order nonlinear models. In addition, we hope to provide some theoretical help for the research in related fields.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29

【相似文献】