【摘要】：在自然科學(xué)、工程技術(shù)領(lǐng)域中存在著大量的非線性現(xiàn)象,非線性科學(xué)也廣泛應(yīng)用于諸如流體力學(xué),光通信等各大領(lǐng)域,因此,對(duì)非線性科學(xué)相關(guān)理論的研究,一直是學(xué)術(shù)界的熱點(diǎn)之一。孤子理論,作為非線性科學(xué)的重要分支,也吸引了科學(xué)家們大量的關(guān)注。對(duì)于非線性模型的解析性質(zhì)研究,求出相關(guān)方程的孤子解是至關(guān)重要的一環(huán),許多求非線性發(fā)展方程孤子解的方法也已被提出。本文通過Hirota雙線性方法,Bell多項(xiàng)式法,B(?)cklund變換法研究了一些非線性模型的孤子解,同時(shí)對(duì)求出的解進(jìn)行了一些解析性質(zhì)研究,如Lax對(duì),無窮守恒律。此外,計(jì)算機(jī)符號(hào)計(jì)算是對(duì)非線性模型求解研究的重要工具。本文的主要內(nèi)容可以分為如下五個(gè)部分:第一章緒論介紹了本文的研究背景及研究現(xiàn)狀,包括孤子理論的發(fā)展歷史及發(fā)展現(xiàn)狀,符號(hào)計(jì)算的基本知識(shí)。第二章介紹了本文研究非線性發(fā)展方程的解析性質(zhì)所使用的方法—-Hirota雙線性方法,Bell多項(xiàng)式法,B(?)cklund變換法。包括方法的理論基礎(chǔ)及具體步驟。第三章研究了雙波形二階Korteweg-de Vries (TKdV)方程,首先引入一個(gè)輔助變量,進(jìn)而通過Bell多項(xiàng)式法,Hirota雙線性方法,B(?)cklund變換法和符號(hào)計(jì)算求出了方程的雙線性形式和B(?)cklund變換,計(jì)算出了方程的N孤子解,并通過作圖,分析了孤子傳播和碰撞的特征,得出了 TKdV方程多孤子之間發(fā)生彈性碰撞的結(jié)論。第四章研究了變系數(shù)modified Kadomtsev-Petviashvili(mKP)方程,通過輔助函數(shù)的引入,利用Bell多項(xiàng)式法,Hirota雙線性方法,符號(hào)計(jì)算,求出了方程的多孤子解及B(?)cklund變換,根據(jù)孤子解的形式,利用Mathematica軟件作圖,分析了孤子解描述沖擊波,鐘形孤立波,倒鐘形孤立波的傳播性質(zhì),產(chǎn)生條件,以及變系數(shù)對(duì)波的傳播的影響。三種波之間的彈性與非彈性碰撞也在本章中被討論。第五章介紹了 Lax對(duì)及無窮守恒律的理論背景及研究意義,以3+1維Jimbo-Mi wa方程為研究對(duì)象,求出了該方程的Bell多項(xiàng)式形式的B(?)cklund變換(BT),基于此BT,推導(dǎo)出了3+1維JM方程的Lax系統(tǒng)以及無窮多個(gè)守恒律。第六章是全文的結(jié)束語,對(duì)全文中的研究工作作了總結(jié),也對(duì)研究過程中遇到的問題,未來的研究方向作出了展望。
[Abstract]:There are a lot of nonlinear phenomena in the field of natural science and engineering technology, and the nonlinear science is also widely used in many fields such as fluid mechanics, optical communication and so on. Therefore, the research on the related theories of nonlinear science is carried out. It has always been one of the hot spots in academia. Soliton theory, as an important branch of nonlinear science, also attracts scientists' attention. For the study of the analytical properties of nonlinear models, it is very important to find the soliton solutions of related equations, and many methods for finding soliton solutions of nonlinear evolution equations have also been proposed. In this paper, the soliton solutions of some nonlinear models are studied by means of Hirota bilinear method and Bell polynomial method, B (?) cklund transform. At the same time, some analytical properties of the obtained solutions, such as Lax pairs and infinite conservation laws, are studied. In addition, computer symbolic computation is an important tool for solving nonlinear models. The main contents of this paper can be divided into the following five parts: the first chapter introduces the research background and research status of this paper, including the development history and current situation of soliton theory, the basic knowledge of symbol calculation. In chapter 2, the Hirota bilinear method, the Bell polynomial method, B (? cklund transform method are introduced to study the analytical properties of nonlinear evolution equations. Including the theoretical basis of the method and specific steps. In the third chapter, the second order Korteweg-de Vries (TKdV) equation with double waveforms is studied. Firstly, an auxiliary variable is introduced, and then the Bell polynomial method and the Hirota bilinear method are used. The bilinear form of the equation and the B (?) cklund transform are obtained by B (?) cklund transformation method and symbolic calculation. The N soliton solution of the equation is calculated, and the characteristics of soliton propagation and collision are analyzed by drawing. The elastic collision between the multi-solitons of the TKdV equation is obtained. In chapter 4, the variable coefficient modified Kadomtsev-Petviashvili (mKP) equation is studied. By introducing auxiliary function, using Bell polynomial method, Hirota bilinear method and symbolic calculation, the multi-soliton solution and B (?) cklund transformation of the equation are obtained, according to the form of soliton solution. The propagation properties of shock wave bell solitary wave inverted bell solitary wave and the effect of variable coefficient on the propagation of shock wave are analyzed by means of the Mathematica software. The results show that the soliton solution can be used to describe the propagation of shock wave bell-shaped solitary wave and inverted bell-shaped solitary wave. Elastic and inelastic collisions between the three waves are also discussed in this chapter. In chapter 5, the theoretical background and research significance of Lax pair and infinite conservation law are introduced. Taking 31-dimensional Jimbo-Mi wa equation as the research object, the B (?) cklund transform (BT), in the form of Bell polynomials of the equation is obtained. Based on this BT, the B (?) cklund transformation of the equation is obtained. The Lax system of the 3 1 dimensional JM equation and infinitely many conservation laws are derived. The sixth chapter is the conclusion of the thesis, which summarizes the research work in this paper, and looks forward to the problems encountered in the research process and the future research direction.
【學(xué)位授予單位】：北京郵電大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O175

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