基于符号计算的非线性发展方程的孤子解及其相关的性质

发布时间：2018-02-09 20:27

本文关键词： 非线性发展方程孤子解 Hirota双线性方法 Bell多项式方法弹性碰撞非弹性碰撞　出处：《北京邮电大学》2015年硕士论文　论文类型：学位论文

【摘要】：随着数学和物理学等学科的发展,对于线性系统,人们已经有了很深入的了解及应用,但是线性系统只是针对复杂现象的近似线性抽象,在自然科学和工程技术中,大多数现象、模型是不能仅用线性系统来描述的,这就激发了人们更加深入研究其本质的动力和兴趣,进而使得非线性科学得以产生并迅速发展。因为非线性因素在大多数的自然科学领域乃至社会科学领域中都会遇到,而且相对于线性系统,非线性系统更能准确的描述各种复杂现象。因此,非线性科学在理论研究与应用中发挥着越来越重要的作用,在各个领域迅速发展,如流体力学、光纤通信、等离子体物理等。孤子,作为非线性科学的一个分支,可以有效的描述非线性中相关波的传播、碰撞、能量转换等现象,近来,在自然科学等领域里,得到了长足的发展,受到了大量关注,而求出孤子解是首要的基础、如今,得到孤子解的方法有很多种,如：Hirota双线性方法,Backlund变换,Bell多项式方法,Darboux变换法等。本文主要是结合了几种方法,利用渐近分析、作图,解析的研究了一些非线性发展方程的孤子解及解的性质,主要工作如下：(1)解析研究了一个描述离子声波在带有非等温电子的等离子体中的传播的扩展修正的Kadomtsev-Petviashvili方程。基于Bell多项式的性质,得到了其双线性形式Backlund变换,并结合Hirota双线性方法,求出了方程的孤子解。图像分析了孤子的传播和孤子之间的碰撞性质：给出了产生钟形孤子、倒钟形孤子和冲击波所需要的参数条件,讨论了参数对孤子传播的影响,并分别展示了不同类型的弹性碰撞和非弹性碰撞。(2)研究了Jaulent-Miodek梯队中的一个2+l维非线性水波方程。通过Bell多项式、Hirota方法和符号计算,得出了其双线性方程不Backlund变换,并推导出了N孤子解,讨论了孤子的传播和相互作用特征,给出了发生平行弹性碰撞的条件。(3)对于一个描述反应的混合物及浅水波的3+1维破裂孤子方程,求出了其单冲击波解、双冲击波解、三冲击波解和N冲击波解。根据求得的解,通过图像分析了冲击波在不同平面上的传播和相互作用的性质,得到了斜弹性碰撞和平行弹性碰撞。(4)考虑正交极化光波在各向同性光纤中的传播,本文研究了相干耦合非线性Schrodinger方程。借助Hirota方法和符号计算,得到了方程的双线性形式,构造了其明单孤子解和明双孤子解。通过渐近分析,讨论了孤子的传播特性和孤子之间的相互作用,展示了迎面弹性碰撞和追赶弹性碰撞各自的图像,并讨论了参数对孤子振幅的影响。(5)针对描述双折射光纤的3+1维耦合非线性Schrodinger方程,借助Hirota方法和符号计算,并通过引入辅助函数,得到了方程的双线性形式,构造了其明单孤子解和明双孤子解,解析的研究了孤子的传播和孤子之间相互作用的性质：得到了单峰孤子和双峰孤子,分析了孤子之间弹性碰撞和非弹性碰撞的不同性质。
[Abstract]:With the development of mathematics and physics and other disciplines, for linear systems, people have a deep understanding and application, but the linear system just for the complicated phenomenon of approximate linear abstract, most phenomena in natural science and engineering technology, and the model is not only used to describe the linear system, this makes people more the in-depth study of the nature of power and interests, thus making the cause and development of nonlinear science. Because of the nonlinear factors encountered in most areas of natural science and social science, and compared with the linear system, nonlinear system can more accurately describe the complex phenomena. Therefore, nonlinear science plays an increasingly important role in the theoretical research and application, the rapid development in various fields, such as fluid mechanics, plasma physics, optical fiber communication, etc. as a non line soliton. A branch of science, can describe the nonlinear wave propagation, effective collision, energy conversion and other phenomena, recently, in natural science and other fields, has got considerable development, has been a lot of attention, and obtained the soliton solution is the primary foundation of today, to the soliton solution method many, such as: Hirota bilinear method, Backlund transform, Bell polynomial method, Darboux transform method. This paper is a combination of several methods, using asymptotic analysis, mapping, analytic study of some soliton solutions of nonlinear evolution equations and solutions, the main work is as follows: (1) analysis of the extended Kadomtsev-Petviashvili equation a description of the ion acoustic wave in plasma with non isothermal electron propagation in the modification. The properties of the Bell polynomials based on the bilinear form of Backlund transform, and Hirota bilinear square Method, obtained the equation of soliton solutions. The image analysis of collision between solitons and soliton propagation properties: given the bell soliton parameters need to inverse bell solitons and shock waves. The influence of parameters on the propagation of solitons are discussed respectively, and shows the different types of elastic collision and non elastic collision. (2) Jaulent-Miodek was studied in an echelon 2+l dimensional nonlinear wave equation. By using the Bell polynomial, Hirota method and symbolic computation, the bilinear equations Backlund transform, and deduced the N soliton solution, discussed the soliton propagation and interaction characteristics, given parallel elastic collision conditions (. 3) for a description of the reaction mixture and the shallow water wave 3+1 dimensional breaking soliton equation, the wave solution of single shock wave solutions, double impact, three shock wave solutions and N shock wave solutions. According to the solution, through the image The nature of the shock wave propagation in different planes and interaction, has been inclined elastic collision and parallel elastic collision. (4) consider the propagation of orthogonal polarization wave in isotropic fiber, studied coherent coupling nonlinear Schrodinger equation. By means of Hirota method and symbolic computation, the bilinear form equation. Construct the Ming single soliton and double soliton solutions. Through asymptotic analysis, discuss the interaction between solitons and soliton propagation, show head-on elastic collision and chasing elastic collision their image, and discusses the influence of parameters on the soliton amplitude. (5) according to the description of birefringence fiber 3+1 dimension coupled nonlinear Schrodinger equation by using Hirota method and symbolic computation, and through the introduction of auxiliary function, the bilinear form equation, we construct the Ming Jie Heming single soliton double soliton solution, analysis The properties of soliton propagation and interaction between solitons are studied: single peak soliton and Shuangfeng soliton are obtained, and the different properties of elastic collision and inelastic collision between solitons are analyzed.

【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

【共引文献】