几类非线性发展方程的孤立子解

发布时间：2018-05-05 21:06

本文选题：非线性发展方程 + 达布变换　；参考：《太原理工大学》2017年硕士论文

【摘要】：本文主要研究了三类高阶非线性发展方程,分别为广义耦合Hirota方程,耦合Hirota方程和高阶非线性薛定谔(NLS)方程.基于达布变换方法,得到了广义耦合Hirota方程的多种孤立子解.同时,系统地探究了耦合Hirota方程和高阶NLS方程的呼吸子-孤子转换机制及非线性波之间的相互作用.全文安排如下:第一章首先介绍了孤子理论的主要内容和研究现状,其次阐述了孤子理论中达布变换方法的基本思想,最后简述本文主要工作.第二章研究了一类广义耦合Hirota方程,在方程中同时考虑高阶非线性项和线性增益(损耗)项.基于达布变换得到方程的周期解,呼吸子解和怪波解,并通过图像分析线性增益(损耗)项和高阶项对孤立子解的传播特性影响.第三章基于达布变换方法研究了耦合Hirota方程,此时方程中不再考虑线性增益(损耗)项.基于平面波背景得到一阶呼吸子表达式,进而建立了呼吸子-孤子的转换机制并得到方程的不同类型局域解和周期解.同时,通过调控参数模拟图形分析不同类型解结构的传播特性和相互作用.第四章基于达布变换方法研究了一类高阶NLS方程,在方程中考虑三次五次非线性项和其它高阶项.通过一阶新解表达式导出方程呼吸子-孤子转换的精确参数关系式,并调控参数分析非线性波之间的相互作用.第五章总结全文并展望未来.
[Abstract]:In this paper, three kinds of higher order nonlinear evolution equations are studied, which are generalized coupled Hirota equation, coupled Hirota equation and high order nonlinear Schrodinger equation. Several soliton solutions of generalized coupled Hirota equation are obtained based on Darboux transform method. At the same time, the mechanism of respiration and soliton conversion and the interaction between nonlinear waves are systematically investigated for coupled Hirota equation and higher order NLS equation. The full text is arranged as follows: in the first chapter, the main contents and research status of soliton theory are introduced, and then the basic idea of Darboux transform method in soliton theory is expounded. Finally, the main work of this paper is briefly described. In chapter 2, we study a class of generalized coupled Hirota equations, in which high order nonlinear terms and linear gain (loss) terms are considered at the same time. Based on Darboux transform, the periodic solution, respiratory solution and odd wave solution of the equation are obtained, and the effects of linear gain (loss) term and higher order term on the propagation characteristics of soliton solution are analyzed by image analysis. In chapter 3, the coupled Hirota equation is studied based on the Darboux transform method, in which the linear gain (loss) term is not considered in the equation. Based on the plane wave background, the expression of the first order respiratory operator is obtained, and the conversion mechanism between the respiratory and soliton is established, and the local solutions and periodic solutions of the equation are obtained. At the same time, the propagation characteristics and interactions of different types of solution structures are analyzed by simulating the parameters. In chapter 4, we study a class of higher order NLS equations based on Darboux transform method. We consider cubic quintic nonlinear terms and other higher order terms in the equation. The exact parametric expression of the respiration soliton transformation of the equation is derived by the expression of the first order new solution, and the control parameters are adjusted to analyze the interaction between nonlinear waves. Chapter five summarizes the full text and looks forward to the future.
【学位授予单位】：太原理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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