幾類孤子方程的可積性研究
發(fā)布時間：2019-04-09 19:22
【摘要】：本文借助Bell多項式方法、Riemann theta函數(shù)周期波解方法、李對稱分析方法從不同的角度研究一些重要的孤子方程的可積性問題,其中包括:精確解、B¨acklund變換、李對稱、守恒律等.第一章,介紹孤立子理論的研究背景,分別介紹了本文采用的三種研究孤子方程可積性方法的研究背景及現(xiàn)狀,最后概括性地介紹本文的選題與主要工作.第二章,用Bell多項式方法探究(3+1)-維非線性演化方程,導出了該方程的雙線性形式、雙線性B¨acklund變換,利用線性疊加原理和同宿測試方法求出了方程的波解和周期孤立波解,并且對解的圖像進行了模擬.第三章,導出了求解Riemann-theta函數(shù)周期波解的方法,并利用此種方法求解Hirota-Satsuma淺水波方程1-周期和2-周期波解,最后對求得的周期波解做漸近分析,證明了參數(shù)在一定限制條件下周期波解趨于孤子解.第四章,利用李對稱分析方法研究了Drinfeld-Sokolov-Wilson系統(tǒng)的無窮小生成元、對稱約化,運用Noether定理導出了參數(shù)為特殊值時的守恒律,另外運用新守恒定理導出了系統(tǒng)對應無窮小生成元的守恒律,由此可見守恒律意義下該系統(tǒng)是可積的.第五章,對本文的研究課題做了總結和進一步的展望.
[Abstract]:In this paper, by means of the Bell polynomial method, Riemann theta function periodic wave solution method, the lie symmetry analysis method is used to study the integrability of some important soliton equations from different angles, including exact solution, B\ + acklund transformation, lie symmetry, conservation law, and so on. In the first chapter, the research background of soliton theory is introduced, and the research background and present situation of the three methods of soliton equation integrability are introduced respectively. Finally, the topic selection and main work of this paper are briefly introduced. In the second chapter, the (31)-dimensional nonlinear evolution equation is investigated by using the Bell polynomial method. The bilinear form of the equation, bilinear B\ + acklund transformation, is derived. The wave solution and periodic solitary wave solution of the equation are obtained by means of linear superposition principle and homoclinic test method, and the image of the solution is simulated. In chapter 3, the method of solving the periodic wave solution of Riemann-theta function is derived, and the 1-period and 2-period wave solutions of the Hirota-Satsuma shallow water wave equation are solved by this method. Finally, the asymptotic analysis of the obtained periodic wave solution is made. It is proved that the periodic wave solutions tend to soliton solutions under certain constraints. In chapter 4, the infinitesimal generator and symmetry reduction of Drinfeld-Sokolov-Wilson system are studied by means of lie symmetry analysis, and the conservation laws when parameters are special values are derived by using Noether theorem. In addition, the conservation law of the infinitesimal generator is derived by using the new conservation theory, which shows that the system is integrable in the sense of conservation law. In the fifth chapter, the research topic of this paper is summarized and further prospected.
【學位授予單位】：中國礦業(yè)大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O175.29
本文編號：2455467
[Abstract]:In this paper, by means of the Bell polynomial method, Riemann theta function periodic wave solution method, the lie symmetry analysis method is used to study the integrability of some important soliton equations from different angles, including exact solution, B\ + acklund transformation, lie symmetry, conservation law, and so on. In the first chapter, the research background of soliton theory is introduced, and the research background and present situation of the three methods of soliton equation integrability are introduced respectively. Finally, the topic selection and main work of this paper are briefly introduced. In the second chapter, the (31)-dimensional nonlinear evolution equation is investigated by using the Bell polynomial method. The bilinear form of the equation, bilinear B\ + acklund transformation, is derived. The wave solution and periodic solitary wave solution of the equation are obtained by means of linear superposition principle and homoclinic test method, and the image of the solution is simulated. In chapter 3, the method of solving the periodic wave solution of Riemann-theta function is derived, and the 1-period and 2-period wave solutions of the Hirota-Satsuma shallow water wave equation are solved by this method. Finally, the asymptotic analysis of the obtained periodic wave solution is made. It is proved that the periodic wave solutions tend to soliton solutions under certain constraints. In chapter 4, the infinitesimal generator and symmetry reduction of Drinfeld-Sokolov-Wilson system are studied by means of lie symmetry analysis, and the conservation laws when parameters are special values are derived by using Noether theorem. In addition, the conservation law of the infinitesimal generator is derived by using the new conservation theory, which shows that the system is integrable in the sense of conservation law. In the fifth chapter, the research topic of this paper is summarized and further prospected.
【學位授予單位】：中國礦業(yè)大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O175.29
【參考文獻】
相關博士學位論文 前1條
1 王云虎;基于符號計算的可積系統(tǒng)的若干問題研究[D];華東師范大學;2013年
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