计算机符号计算在若干非线性模型求解中的应用
发布时间：2019-02-26 09:53
【摘要】：在自然科学、工程技术领域中存在着大量的非线性现象,非线性科学也广泛应用于诸如流体力学,光通信等各大领域,因此,对非线性科学相关理论的研究,一直是学术界的热点之一。孤子理论,作为非线性科学的重要分支,也吸引了科学家们大量的关注。对于非线性模型的解析性质研究,求出相关方程的孤子解是至关重要的一环,许多求非线性发展方程孤子解的方法也已被提出。本文通过Hirota双线性方法,Bell多项式法,B(?)cklund变换法研究了一些非线性模型的孤子解,同时对求出的解进行了一些解析性质研究,如Lax对,无穷守恒律。此外,计算机符号计算是对非线性模型求解研究的重要工具。本文的主要内容可以分为如下五个部分:第一章绪论介绍了本文的研究背景及研究现状,包括孤子理论的发展历史及发展现状,符号计算的基本知识。第二章介绍了本文研究非线性发展方程的解析性质所使用的方法—-Hirota双线性方法,Bell多项式法,B(?)cklund变换法。包括方法的理论基础及具体步骤。第三章研究了双波形二阶Korteweg-de Vries (TKdV)方程,首先引入一个辅助变量,进而通过Bell多项式法,Hirota双线性方法,B(?)cklund变换法和符号计算求出了方程的双线性形式和B(?)cklund变换,计算出了方程的N孤子解,并通过作图,分析了孤子传播和碰撞的特征,得出了 TKdV方程多孤子之间发生弹性碰撞的结论。第四章研究了变系数modified Kadomtsev-Petviashvili(mKP)方程,通过辅助函数的引入,利用Bell多项式法,Hirota双线性方法,符号计算,求出了方程的多孤子解及B(?)cklund变换,根据孤子解的形式,利用Mathematica软件作图,分析了孤子解描述冲击波,钟形孤立波,倒钟形孤立波的传播性质,产生条件,以及变系数对波的传播的影响。三种波之间的弹性与非弹性碰撞也在本章中被讨论。第五章介绍了 Lax对及无穷守恒律的理论背景及研究意义,以3+1维Jimbo-Mi wa方程为研究对象,求出了该方程的Bell多项式形式的B(?)cklund变换(BT),基于此BT,推导出了3+1维JM方程的Lax系统以及无穷多个守恒律。第六章是全文的结束语,对全文中的研究工作作了总结,也对研究过程中遇到的问题,未来的研究方向作出了展望。
[Abstract]:There are a lot of nonlinear phenomena in the field of natural science and engineering technology, and the nonlinear science is also widely used in many fields such as fluid mechanics, optical communication and so on. Therefore, the research on the related theories of nonlinear science is carried out. It has always been one of the hot spots in academia. Soliton theory, as an important branch of nonlinear science, also attracts scientists' attention. For the study of the analytical properties of nonlinear models, it is very important to find the soliton solutions of related equations, and many methods for finding soliton solutions of nonlinear evolution equations have also been proposed. In this paper, the soliton solutions of some nonlinear models are studied by means of Hirota bilinear method and Bell polynomial method, B (?) cklund transform. At the same time, some analytical properties of the obtained solutions, such as Lax pairs and infinite conservation laws, are studied. In addition, computer symbolic computation is an important tool for solving nonlinear models. The main contents of this paper can be divided into the following five parts: the first chapter introduces the research background and research status of this paper, including the development history and current situation of soliton theory, the basic knowledge of symbol calculation. In chapter 2, the Hirota bilinear method, the Bell polynomial method, B (? cklund transform method are introduced to study the analytical properties of nonlinear evolution equations. Including the theoretical basis of the method and specific steps. In the third chapter, the second order Korteweg-de Vries (TKdV) equation with double waveforms is studied. Firstly, an auxiliary variable is introduced, and then the Bell polynomial method and the Hirota bilinear method are used. The bilinear form of the equation and the B (?) cklund transform are obtained by B (?) cklund transformation method and symbolic calculation. The N soliton solution of the equation is calculated, and the characteristics of soliton propagation and collision are analyzed by drawing. The elastic collision between the multi-solitons of the TKdV equation is obtained. In chapter 4, the variable coefficient modified Kadomtsev-Petviashvili (mKP) equation is studied. By introducing auxiliary function, using Bell polynomial method, Hirota bilinear method and symbolic calculation, the multi-soliton solution and B (?) cklund transformation of the equation are obtained, according to the form of soliton solution. The propagation properties of shock wave bell solitary wave inverted bell solitary wave and the effect of variable coefficient on the propagation of shock wave are analyzed by means of the Mathematica software. The results show that the soliton solution can be used to describe the propagation of shock wave bell-shaped solitary wave and inverted bell-shaped solitary wave. Elastic and inelastic collisions between the three waves are also discussed in this chapter. In chapter 5, the theoretical background and research significance of Lax pair and infinite conservation law are introduced. Taking 31-dimensional Jimbo-Mi wa equation as the research object, the B (?) cklund transform (BT), in the form of Bell polynomials of the equation is obtained. Based on this BT, the B (?) cklund transformation of the equation is obtained. The Lax system of the 3 1 dimensional JM equation and infinitely many conservation laws are derived. The sixth chapter is the conclusion of the thesis, which summarizes the research work in this paper, and looks forward to the problems encountered in the research process and the future research direction.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
本文编号：2430669
[Abstract]:There are a lot of nonlinear phenomena in the field of natural science and engineering technology, and the nonlinear science is also widely used in many fields such as fluid mechanics, optical communication and so on. Therefore, the research on the related theories of nonlinear science is carried out. It has always been one of the hot spots in academia. Soliton theory, as an important branch of nonlinear science, also attracts scientists' attention. For the study of the analytical properties of nonlinear models, it is very important to find the soliton solutions of related equations, and many methods for finding soliton solutions of nonlinear evolution equations have also been proposed. In this paper, the soliton solutions of some nonlinear models are studied by means of Hirota bilinear method and Bell polynomial method, B (?) cklund transform. At the same time, some analytical properties of the obtained solutions, such as Lax pairs and infinite conservation laws, are studied. In addition, computer symbolic computation is an important tool for solving nonlinear models. The main contents of this paper can be divided into the following five parts: the first chapter introduces the research background and research status of this paper, including the development history and current situation of soliton theory, the basic knowledge of symbol calculation. In chapter 2, the Hirota bilinear method, the Bell polynomial method, B (? cklund transform method are introduced to study the analytical properties of nonlinear evolution equations. Including the theoretical basis of the method and specific steps. In the third chapter, the second order Korteweg-de Vries (TKdV) equation with double waveforms is studied. Firstly, an auxiliary variable is introduced, and then the Bell polynomial method and the Hirota bilinear method are used. The bilinear form of the equation and the B (?) cklund transform are obtained by B (?) cklund transformation method and symbolic calculation. The N soliton solution of the equation is calculated, and the characteristics of soliton propagation and collision are analyzed by drawing. The elastic collision between the multi-solitons of the TKdV equation is obtained. In chapter 4, the variable coefficient modified Kadomtsev-Petviashvili (mKP) equation is studied. By introducing auxiliary function, using Bell polynomial method, Hirota bilinear method and symbolic calculation, the multi-soliton solution and B (?) cklund transformation of the equation are obtained, according to the form of soliton solution. The propagation properties of shock wave bell solitary wave inverted bell solitary wave and the effect of variable coefficient on the propagation of shock wave are analyzed by means of the Mathematica software. The results show that the soliton solution can be used to describe the propagation of shock wave bell-shaped solitary wave and inverted bell-shaped solitary wave. Elastic and inelastic collisions between the three waves are also discussed in this chapter. In chapter 5, the theoretical background and research significance of Lax pair and infinite conservation law are introduced. Taking 31-dimensional Jimbo-Mi wa equation as the research object, the B (?) cklund transform (BT), in the form of Bell polynomials of the equation is obtained. Based on this BT, the B (?) cklund transformation of the equation is obtained. The Lax system of the 3 1 dimensional JM equation and infinitely many conservation laws are derived. The sixth chapter is the conclusion of the thesis, which summarizes the research work in this paper, and looks forward to the problems encountered in the research process and the future research direction.
【学位授予单位】：北京邮电大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
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