等谱与非等谱的广义带导数非线性薛定谔方程的精确解

发布时间：2018-04-05 10:50

本文选题：导数非线性薛定谔方程　切入点：Hirota方法　出处：《东华理工大学》2017年硕士论文

【摘要】：本文研究了等谱与非等谱的广义带导数非线性薛定谔方程的精确解问题。主要内容包括:利用Hirota方法得到广义非等谱的导数非线性薛定谔方程的N-孤子解,给出了解的动力学特征,并将此方程及解约化到非等谱的导数非线性薛定谔方程及解;利用Wronskian技巧得到广义带导数的非线性薛定谔方程的广义双Wronskian解、孤子解及有理解。第一章,主要回顾了孤子理论的产生和发展历程,并介绍了几种孤子方程常见的求解方法。第二章,简单地叙述了双线性导数和Wronskian行列式中的一些基本概念和重要性质。第三章,从Kaup-Newell谱问题出发,导出广义非等谱的导数非线性薛定谔方程,该方程可在合适的条件下,利用Hirota方法,寻找出该方程的单孤子、双孤子解和N-孤子解,给出单孤子解及双孤子相互作用的动力学特征,并通过约化,进一步给出非等谱的导数非线性薛定谔方程Hirota形式的N-孤子解。第四章,在Wronskian技巧的基础之上,将双Wronskian元素满足的条件推广至矩阵形式,从而给出方程的广义双Wronskian解,并进一步得到该方程的孤子解及有理解。第五章,对全文进行总结以及对后续内容的展望。
[Abstract]:In this paper, we study the exact solutions of the generalized nonlinear Schrodinger equation with derivative for isospectral and nonisophoric.The main contents are as follows: the N- soliton solution of the derivative nonlinear Schrodinger equation is obtained by using the Hirota method, the dynamical characteristics of the solution are given, and the equation is reduced to the derivative nonlinear Schrodinger equation and the solution of the non-isospectral nonlinear Schrodinger equation.By using the Wronskian technique, the generalized double Wronskian solution, soliton solution and understanding of the nonlinear Schrodinger equation with derivatives are obtained.In chapter 1, the generation and development of soliton theory are reviewed, and several common soliton equations are introduced.In chapter 2, some basic concepts and important properties of bilinear derivative and Wronskian determinant are briefly described.In chapter 3, from the problem of Kaup-Newell spectrum, the derivative nonlinear Schrodinger equation of generalized non-isospectral spectrum is derived. Under suitable conditions and using Hirota method, the single soliton solution, double soliton solution and N-soliton solution of the equation can be found.The dynamical characteristics of the single soliton solution and the double soliton interaction are given, and the N-soliton solutions in the form of Hirota form of the nonlinear differential Schrodinger equation with non-isospectral derivatives are further given by means of reduction.In chapter 4, on the basis of Wronskian's technique, the condition of double Wronskian element is extended to matrix form, and then the generalized double Wronskian solution of the equation is given, and the soliton solution and the understanding of the equation are obtained.The fifth chapter summarizes the full text and looks forward to the following content.
【学位授予单位】：东华理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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