本文選題：非線性發(fā)展方程 + 達布變換　； 參考：《太原理工大學》2017年碩士論文

【摘要】：本文主要研究了三類高階非線性發(fā)展方程,分別為廣義耦合Hirota方程,耦合Hirota方程和高階非線性薛定諤(NLS)方程.基于達布變換方法,得到了廣義耦合Hirota方程的多種孤立子解.同時,系統(tǒng)地探究了耦合Hirota方程和高階NLS方程的呼吸子-孤子轉(zhuǎn)換機制及非線性波之間的相互作用.全文安排如下:第一章首先介紹了孤子理論的主要內(nèi)容和研究現(xiàn)狀,其次闡述了孤子理論中達布變換方法的基本思想,最后簡述本文主要工作.第二章研究了一類廣義耦合Hirota方程,在方程中同時考慮高階非線性項和線性增益(損耗)項.基于達布變換得到方程的周期解,呼吸子解和怪波解,并通過圖像分析線性增益(損耗)項和高階項對孤立子解的傳播特性影響.第三章基于達布變換方法研究了耦合Hirota方程,此時方程中不再考慮線性增益(損耗)項.基于平面波背景得到一階呼吸子表達式,進而建立了呼吸子-孤子的轉(zhuǎn)換機制并得到方程的不同類型局域解和周期解.同時,通過調(diào)控參數(shù)模擬圖形分析不同類型解結(jié)構(gòu)的傳播特性和相互作用.第四章基于達布變換方法研究了一類高階NLS方程,在方程中考慮三次五次非線性項和其它高階項.通過一階新解表達式導出方程呼吸子-孤子轉(zhuǎn)換的精確參數(shù)關(guān)系式,并調(diào)控參數(shù)分析非線性波之間的相互作用.第五章總結(jié)全文并展望未來.
[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.
【學位授予單位】：太原理工大學
【學位級別】：碩士
【學位授予年份】：2017
【分類號】：O175.29

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本文編號：1849300