本文選題：Hirota雙線性方法 + 非線性演化方程��； 參考：《山東科技大學(xué)》2017年碩士論文

【摘要】：近年來,離散可積系統(tǒng)已經(jīng)被廣泛地應(yīng)用到很多領(lǐng)域,例如光學(xué)、流體力學(xué)、磁流體學(xué)等.相比連續(xù)可積系統(tǒng),離散可積系統(tǒng)可以更好的刻畫自然界物質(zhì)的運(yùn)動規(guī)律,成為近年來研究熱點(diǎn)問題.除此之外,孤子方程有理解的研究可以深刻的描述許多物理現(xiàn)象,具有重要的潛在應(yīng)用價值.在本文中,我們構(gòu)造了一個新的離散譜算子,得到了離散晶格方程的可積辛映射和Hamiltonian結(jié)構(gòu).隨后運(yùn)用Hirota雙線性方法分別得到了兩類非線性發(fā)展方程的有理解、Lump解和怪波解.全文結(jié)構(gòu)如下:1介紹了孤立子理論的產(chǎn)生以及發(fā)展現(xiàn)狀,簡單概述了 Hirota雙線性求解方法的思想和應(yīng)用.2依據(jù)離散可積系統(tǒng)的屠格式理論,構(gòu)造了一個新的離散譜算子,通過離散可積系統(tǒng)的零曲率方程,得到了離散可積系統(tǒng)的Hamiltonian結(jié)構(gòu)并列舉了一組可積耦合方程.隨后借助于Bargmann對稱約束條件,得到了兩組顯示的對稱約束,從而給出了離散可積系統(tǒng)的可積辛映射.最后運(yùn)用對稱理論求解的思想,對離散可積耦合方程進(jìn)行求解研究,并分析了步長對解的動力學(xué)形態(tài)的影響.3根據(jù)雙線性求解方法的思維和符號計算的技巧,首先運(yùn)用多項式方法得到了(3+1)-維類淺水波方程的5組有理解,通過選取合適的參數(shù),展示出有理解的動力學(xué)特征,隨后借助二次函數(shù)思想,得到了約化(3+1)維類淺水波方程的lump解,通過參數(shù)約束確保此lump解在(x,y)平面的任何方向上均是局部有理的.4首先根據(jù)二次函數(shù)理論和雙線性方法得到了約化(3+1)維非線性發(fā)展方程的lump解,更進(jìn)一步地,將二次函數(shù)方法延拓為二次函數(shù)與指數(shù)函數(shù)的結(jié)合,從而得到了lump解和線孤子解的相互作用,由于lump解和線孤子解運(yùn)動方向的不同,展示了兩種動力學(xué)行為,孤子融合和孤子裂變.隨后將二次函數(shù)延拓到二次函數(shù)與雙曲余弦函數(shù)的結(jié)合,得到了二元函數(shù)的怪波解,相比于以前二維函數(shù)的線怪波解,此處得到的怪波解在(x,y)平面上均是局部有理的.需要強(qiáng)調(diào)的是,第四章中,非線性發(fā)展方程的怪波解在(x,y)平面上任何方向均有怪波的特性,將一維怪波推廣到二維怪波,改善了傳統(tǒng)意義上的二維線怪波解.
[Abstract]:In recent years, discrete integrable systems have been widely used in many fields, such as optics, hydrodynamics, magnetohydrology and so on. Compared with continuous integrable systems, discrete integrable systems can better describe the motion laws of natural materials, and have become a hot topic in recent years. In addition, soliton equations can describe many physical phenomena deeply and have important potential application value. In this paper we construct a new discrete spectral operator and obtain the integrable symplectic mapping and Hamiltonian structure of discrete lattice equations. Then, the Hirota bilinear method is used to obtain two kinds of nonlinear evolution equations, which are the understood stump solution and the odd wave solution, respectively. The structure of the paper is as follows: 1. The generation and development of soliton theory are introduced. The idea of Hirota bilinear solution method and its application .2 A new discrete spectral operator are constructed according to the Tu format theory of discrete integrable system. Through the zero curvature equation of discrete integrable system, the Hamiltonian structure of discrete integrable system is obtained and a set of integrable coupling equations are listed. Then, by means of Bargmann's symmetric constraint condition, two sets of shown symmetric constraints are obtained, and then the integrable symplectic mapping of discrete integrable systems is given. Finally, the solution of discrete integrable coupled equations is studied by using the idea of symmetry theory, and the influence of step size on the dynamic shape of solution is analyzed. 3. According to the thinking of bilinear solution method and the technique of symbolic calculation, Firstly, five groups of shallow water wave equations are obtained by using polynomial method. By selecting appropriate parameters, the dynamic characteristics with understanding are shown, and then the idea of quadratic function is used. In this paper, the lump solution of the shallow water wave equation with reduced dimension is obtained. It is ensured that the lump solution is locally rational in any direction of the plane by parameter constraints. At first, according to the quadratic function theory and bilinear method, the lump solution of the reduced 31) dimensional nonlinear evolution equation is obtained. The quadratic function method is extended to the combination of quadratic function and exponential function, and the interaction between the lump solution and the linear soliton solution is obtained. Due to the difference of the motion direction between the lump solution and the linear soliton solution, two dynamic behaviors are shown. Soliton fusion and soliton fission. Then, by combining the quadratic function with the hyperbolic cosine function, the singular wave solution of the binary function is obtained. Compared with the linear anomalous wave solution of the previous two-dimensional function, the singular wave solution obtained here is locally rational in the plane of the hyperbolic cosine function. It should be emphasized that in chapter 4, the solution of nonlinear evolution equation has the characteristic of strange wave in any direction of the plane. The one-dimensional strange wave is extended to the two-dimensional strange wave, which improves the traditional 2-D linear strange wave solution.
【學(xué)位授予單位】：山東科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O241.7

【參考文獻(xiàn)】

相關(guān)期刊論文 前4條

1 張大軍;寧同科;;可積系統(tǒng)的守恒律[J];上海大學(xué)學(xué)報(自然科學(xué)版);2006年01期

2 曾云波，李翊神;從無窮維可積Hamilton系統(tǒng)到有限維可積Hamilton系統(tǒng)的一個一般途徑[J];數(shù)學(xué)進(jìn)展;1995年02期

3 曹策問,耿獻(xiàn)國;Bargmann系統(tǒng)與耦合Harry-Dym方程解的對合表示[J];數(shù)學(xué)學(xué)報;1992年03期

4 曹策問;AKNS族的Lax方程組的非線性化[J];中國科學(xué)(A輯 數(shù)學(xué) 物理學(xué) 天文學(xué) 技術(shù)科學(xué));1989年07期

相關(guān)博士學(xué)位論文 前2條

1 夏鐵成;吳方法及其在偏微分方程中的應(yīng)用[D];大連理工大學(xué);2002年

2 閆振亞;非線性波與可積系統(tǒng)[D];大連理工大學(xué);2002年

相關(guān)碩士學(xué)位論文 前1條

1 李柱;非線性發(fā)展方程的精確解與可積系統(tǒng)相關(guān)問題的研究[D];山東科技大學(xué);2007年

，

本文編號：2002440