几类孤子方程的可积性研究
发布时间：2019-04-09 19:22
【摘要】：本文借助Bell多项式方法、Riemann theta函数周期波解方法、李对称分析方法从不同的角度研究一些重要的孤子方程的可积性问题,其中包括:精确解、B¨acklund变换、李对称、守恒律等.第一章,介绍孤立子理论的研究背景,分别介绍了本文采用的三种研究孤子方程可积性方法的研究背景及现状,最后概括性地介绍本文的选题与主要工作.第二章,用Bell多项式方法探究(3+1)-维非线性演化方程,导出了该方程的双线性形式、双线性B¨acklund变换,利用线性叠加原理和同宿测试方法求出了方程的波解和周期孤立波解,并且对解的图像进行了模拟.第三章,导出了求解Riemann-theta函数周期波解的方法,并利用此种方法求解Hirota-Satsuma浅水波方程1-周期和2-周期波解,最后对求得的周期波解做渐近分析,证明了参数在一定限制条件下周期波解趋于孤子解.第四章,利用李对称分析方法研究了Drinfeld-Sokolov-Wilson系统的无穷小生成元、对称约化,运用Noether定理导出了参数为特殊值时的守恒律,另外运用新守恒定理导出了系统对应无穷小生成元的守恒律,由此可见守恒律意义下该系统是可积的.第五章,对本文的研究课题做了总结和进一步的展望.
[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.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29
本文编号：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.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175.29
【参考文献】
相关博士学位论文 前1条
1 王云虎;基于符号计算的可积系统的若干问题研究[D];华东师范大学;2013年
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