本文選題：導(dǎo)數(shù)非線性薛定諤方程　切入點(diǎn)：Hirota方法　出處：《東華理工大學(xué)》2017年碩士論文

【摘要】：本文研究了等譜與非等譜的廣義帶導(dǎo)數(shù)非線性薛定諤方程的精確解問題。主要內(nèi)容包括:利用Hirota方法得到廣義非等譜的導(dǎo)數(shù)非線性薛定諤方程的N-孤子解,給出了解的動力學(xué)特征,并將此方程及解約化到非等譜的導(dǎo)數(shù)非線性薛定諤方程及解;利用Wronskian技巧得到廣義帶導(dǎo)數(shù)的非線性薛定諤方程的廣義雙Wronskian解、孤子解及有理解。第一章,主要回顧了孤子理論的產(chǎn)生和發(fā)展歷程,并介紹了幾種孤子方程常見的求解方法。第二章,簡單地?cái)⑹隽穗p線性導(dǎo)數(shù)和Wronskian行列式中的一些基本概念和重要性質(zhì)。第三章,從Kaup-Newell譜問題出發(fā),導(dǎo)出廣義非等譜的導(dǎo)數(shù)非線性薛定諤方程,該方程可在合適的條件下,利用Hirota方法,尋找出該方程的單孤子、雙孤子解和N-孤子解,給出單孤子解及雙孤子相互作用的動力學(xué)特征,并通過約化,進(jìn)一步給出非等譜的導(dǎo)數(shù)非線性薛定諤方程Hirota形式的N-孤子解。第四章,在Wronskian技巧的基礎(chǔ)之上,將雙Wronskian元素滿足的條件推廣至矩陣形式,從而給出方程的廣義雙Wronskian解,并進(jìn)一步得到該方程的孤子解及有理解。第五章,對全文進(jìn)行總結(jié)以及對后續(xù)內(nèi)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.
【學(xué)位授予單位】：東華理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.29

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