基于广义双线性算子的高维浅水波方程的有理解

发布时间：2018-05-02 04:03

本文选题：孤立子理论 + Hirota双线性方法　；参考：《山东科技大学》2017年硕士论文

【摘要】：在非线性科学中,孤立子理论是非常重要的一个分支.孤立子理论模型在物理海洋、量子力学、大气科学、流体力学等领域中用于刻画物理现象,如大气中的阻塞现象(mKdV模型),光纤通讯中的光孤子(KdV模型),金融工程中的金融孤子(非线性Schrodinger方程).因此,对孤子方程的可积性质及其解的研究可以促进诸多领域的发展,是有重大的理论意义和潜在的应用价值.本文主要介绍了 Hirota双线性算子,并利用Hirota双线性方法得到了(2+1)维浅水波方程及(2+1)维类浅水波方程的有理解,通过对有理解的讨论及其图像的分析得出其在实际应用中的价值.全文结构如下:第一章简要介绍了孤立子理论的发现发展过程,概述了孤子方程求解过程中常用的反散射法,Backlund变换和Darboux变换,广田双线性法等.第二章第一部分主要介绍了 Hirota双线性方法的基本思想,并举例介绍了将非线性的偏微分方程转化为线性形式最为典型的三种变换方法:有理变换、对数变换、双对数变换.第二部分对Bell多项式做了简单介绍.第三章第一部分给出双线性算子,通过变量变换得到(2+1)维浅水波方程的双线性形式,然后经过符号计算系数变换等得到该(2+1)浅水波方程的有理解.第二部分给出双线性算子的性质,继而将广义双线性算子与经典双线性算子进行比较,然后通过变量变换得到(2+1)维类浅水波方程的广义双线性形式,并利用Maple符号计算与系数变换得到该方程的有理解.
[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.
【学位授予单位】：山东科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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