本文選題：非線性發(fā)展方程 + 精確解�。� 參考：《內(nèi)蒙古師范大學(xué)》2017年碩士論文

【摘要】：非線性發(fā)展方程(組)的精確求解問(wèn)題是孤立子理論中的重要研究方向之一,它包括提出新的精確求解方法與利用已經(jīng)建立的方法給出非線性發(fā)展方程新的精確解等兩個(gè)方面.由于非線性發(fā)展方程(組)的復(fù)雜性,建立一個(gè)統(tǒng)一的精確求解方法看來(lái)十分困難.但是針對(duì)非線性發(fā)展方程的個(gè)性特征,探索并找出相應(yīng)的較為系統(tǒng)的精確求解的新方法是可以實(shí)現(xiàn)的,一旦建立了一種新的求解方法就有可能獲得所研究的非線性發(fā)展方程的新解或特殊解且它們恰好能解釋新的物理現(xiàn)象.利用已經(jīng)建立的方法求解非線性發(fā)展方程(組)而獲取新解也是精確求解所采用的重要手段,且因方便、快捷、能夠解釋新發(fā)現(xiàn)的物理現(xiàn)象而受到物理學(xué)家、工程技術(shù)人員的關(guān)注.因此,非線性發(fā)展方程的精確求解問(wèn)題的研究具有推動(dòng)求解理論和求解方法的建立,為實(shí)際問(wèn)題的解決和解釋提供有效工具.本文繼許多專家、學(xué)者研究工作的基礎(chǔ)上,利用exp(-φ(ζ))-展開(kāi)法、G'/G-展開(kāi)法和廣義輔助方程法研究一些非線性發(fā)展方程并嘗試構(gòu)造其精確解.本文主要由四部分組成,具體安排如下:第一章簡(jiǎn)要概述孤立子理論的研究意義、輔助方程方法的應(yīng)用狀況以及本文的主要研究工作.第二章簡(jiǎn)要介紹exp(-φ(ζ))-展開(kāi)法,并將其應(yīng)用到(2+1)維耗散長(zhǎng)水波方程、廣義的變系數(shù)KdV-mKdV方程以及變系數(shù)(2+1)維Broer-Kaup方程中,獲得了新的奇異行波解.將本文結(jié)果與其他文獻(xiàn)中用不同方法給出的精確解進(jìn)行比較,得出它們只是本文所給出的解的特殊情形.第三章簡(jiǎn)要介紹G'/G-展開(kāi)法,并以常系數(shù)Newell方程、變系數(shù)Novikov-Veselov方程以及離散復(fù)立方-五次Ginzburg-Landau方程為例,求解獲得含有自由參數(shù)的通解.由此說(shuō)明了 G'/G-展開(kāi)法可以用于求解變系數(shù)方程、復(fù)方程和離散方程等各種類型的非線性發(fā)展方程的求解.G'/G-展開(kāi)法求解過(guò)程簡(jiǎn)單、直接,而且由于自由參數(shù)的任意性,所得的精確解也更加豐富.第四章介紹和分析廣義輔助方程法,通過(guò)構(gòu)造出適當(dāng)?shù)慕獾男问饺デ蠼?2+1)維 Calogero-Bogoyavlenskii-Schiff 方程及變系數(shù)組合 KdV方程獲得更豐富的精確類孤子解.
[Abstract]:The exact solution of nonlinear evolution equations (systems) is one of the important research directions in the soliton theory. It includes two aspects: a new exact solution method and a new exact solution of the nonlinear evolution equation by using the established method. Because of the complexity of nonlinear evolution equations, it is very difficult to establish a unified exact solution method. However, in view of the personality characteristics of nonlinear evolution equations, it is possible to explore and find out a new method to solve the nonlinear evolution equations accurately and systematically. Once a new method is established, it is possible to obtain new solutions or special solutions of the nonlinear evolution equations studied and they can explain the new physical phenomena. Using the established method to solve nonlinear evolution equations (systems) and obtaining new solutions is also an important means of exact solution. It is also a physicist who can explain the newly discovered physical phenomena because of its convenience and rapidity. The attention of engineers and technicians. Therefore, the research on the exact solution of nonlinear evolution equations has promoted the establishment of solving theory and method, and provided an effective tool for the solution and interpretation of practical problems. In this paper, based on the research work of many experts and scholars, some nonlinear evolution equations are studied and their exact solutions are constructed by using expan- 蠁 (味 ~ (-) -expansion method) and generalized auxiliary equation method. This paper is composed of four parts. The main contents are as follows: in chapter one, the research significance of soliton theory, the application of auxiliary equation method and the main research work of this paper are briefly summarized. In chapter 2, expan- 蠁 (味 ~ + -expansion method) is briefly introduced and applied to the dissipative long water wave equation, the generalized variable coefficient KdV-mKdV equation and the variable coefficient 21) -dimensional Broer-Kaup equation. A new singular traveling wave solution is obtained by applying the method to the dissipative long water wave equation in ~ (21) D, the generalized variable coefficient KdV-mKdV equation and the variable coefficient ~ (21) -dimensional Broer-Kaup equation. By comparing the results of this paper with the exact solutions given by different methods in other literatures, it is found that they are only the special cases of the solutions given in this paper. In chapter 3, we briefly introduce the Gon / G- expansion method, and take the Newell equation with constant coefficients, the Novikov-Veselov equation with variable coefficients and the Ginzburg-Landau equation of discrete complex cubic order as examples to obtain the general solution with free parameters. It is shown that the GG / G- expansion method can be used to solve all kinds of nonlinear evolution equations, such as variable coefficient equation, compound equation and discrete equation. The process of solving the nonlinear evolution equation is simple and direct, and because of the arbitrariness of free parameter, the method can be used to solve all kinds of nonlinear evolution equations, such as variable coefficient equation, compound equation and discrete equation. The exact solutions obtained are also more abundant. In chapter 4, the generalized auxiliary equation method is introduced and analyzed. By constructing an appropriate solution form to solve the Calogero-Bogoyavlenskii-Schiff equation and the variable coefficient combined KDV equation, the exact soliton-like solutions are obtained.
【學(xué)位授予單位】：內(nèi)蒙古師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175.29

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