本文選題：孤立子理論 + Hirota雙線性方法　； 參考：《山東科技大學(xué)》2017年碩士論文

【摘要】：在非線性科學(xué)中,孤立子理論是非常重要的一個(gè)分支.孤立子理論模型在物理海洋、量子力學(xué)、大氣科學(xué)、流體力學(xué)等領(lǐng)域中用于刻畫物理現(xiàn)象,如大氣中的阻塞現(xiàn)象(mKdV模型),光纖通訊中的光孤子(KdV模型),金融工程中的金融孤子(非線性Schrodinger方程).因此,對(duì)孤子方程的可積性質(zhì)及其解的研究可以促進(jìn)諸多領(lǐng)域的發(fā)展,是有重大的理論意義和潛在的應(yīng)用價(jià)值.本文主要介紹了 Hirota雙線性算子,并利用Hirota雙線性方法得到了(2+1)維淺水波方程及(2+1)維類淺水波方程的有理解,通過對(duì)有理解的討論及其圖像的分析得出其在實(shí)際應(yīng)用中的價(jià)值.全文結(jié)構(gòu)如下:第一章簡要介紹了孤立子理論的發(fā)現(xiàn)發(fā)展過程,概述了孤子方程求解過程中常用的反散射法,Backlund變換和Darboux變換,廣田雙線性法等.第二章第一部分主要介紹了 Hirota雙線性方法的基本思想,并舉例介紹了將非線性的偏微分方程轉(zhuǎn)化為線性形式最為典型的三種變換方法:有理變換、對(duì)數(shù)變換、雙對(duì)數(shù)變換.第二部分對(duì)Bell多項(xiàng)式做了簡單介紹.第三章第一部分給出雙線性算子,通過變量變換得到(2+1)維淺水波方程的雙線性形式,然后經(jīng)過符號(hào)計(jì)算系數(shù)變換等得到該(2+1)淺水波方程的有理解.第二部分給出雙線性算子的性質(zhì),繼而將廣義雙線性算子與經(jīng)典雙線性算子進(jìn)行比較,然后通過變量變換得到(2+1)維類淺水波方程的廣義雙線性形式,并利用Maple符號(hào)計(jì)算與系數(shù)變換得到該方程的有理解.
[Abstract]:Soliton theory is a very important branch of nonlinear science. The soliton theory model is used to describe physical phenomena in the fields of physical ocean, quantum mechanics, atmospheric science, fluid mechanics, etc. For example, the blocking phenomenon in the atmosphere is mKdV model, the optical soliton in optical fiber communication is KDV model, and the financial soliton in financial engineering (nonlinear Schrodinger equation). Therefore, the study of integrable properties and solutions of soliton equations can promote the development of many fields, which is of great theoretical significance and potential application value. In this paper, the Hirota bilinear operator is introduced, and the Hirota bilinear method is used to get the understanding of the shallow water wave equation and the shallow water wave equation. The value of understanding discussion and image analysis in practical application is obtained. The structure of the paper is as follows: in Chapter 1, the discovery and development of soliton theory are briefly introduced. The backscattering methods such as Backlund transform and Darboux transform, and Kuantian bilinear method are summarized. The first part of chapter two mainly introduces the basic idea of Hirota bilinear method, and introduces three typical transformation methods: rational transformation, logarithmic transformation and double logarithmic transformation, which convert nonlinear partial differential equation into linear form. The second part gives a brief introduction to Bell polynomials. In chapter 3, the bilinear operator is given, the bilinear form of the shallow water wave equation is obtained by variable transformation, and the understanding of the shallow water wave equation is obtained by symbolic coefficient transformation. In the second part, the properties of bilinear operator are given, and then the generalized bilinear operator is compared with the classical bilinear operator, and then the generalized bilinear form of the shallow water wave equation is obtained by variable transformation. The Maple symbolic calculation and coefficient transformation are used to obtain the understanding of the equation.
【學(xué)位授予單位】：山東科技大學(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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