几类非线性方程的可积性研究

发布时间：2018-05-15 09:25

本文选题：可积性 + Bell多项式　；参考：《中国矿业大学》2016年硕士论文

【摘要】：在非线性科学中,孤子理论是一个重要的组成部分,对自然科学和工程技术的发展具有重要意义。在这篇文章中我们主要研究非线性变系数方程的可积性、扩展KP方程的周期波解与可积性、以及(2+1)-维演化方程族的哈密顿结构。在绪论中我们主要介绍孤子理论的产生以及发展状况和研究现状,然后介绍本文的选题和主要工作。第二章,首先根据Bell多项式与Hirota双线性D算子之间的关系推导出变系数方程的双线性导数形式和双线性B?cklund变换,然后我们根据所得到的双线性B?cklund变换,Bell多项式与Hopf-Cole变换v=lnψ之间的关系,导出非线性方程的Lax对和共轭Lax对的具体表现形式,最后我们推导出方程的无穷守恒律。第三章,利用Riemann-theta函数周期波解理论,我们求出扩展KP方程的周期波解,并对所得的周期波解做渐近分析,最后求的方程的一些可积性质。第四章,利用李代数和屠格式的方法产生新的可积方程族,然后对其约化获得它们的哈密顿结构。第五章,对本文进行总结。
[Abstract]:In nonlinear science, soliton theory is an important part of the development of natural science and engineering technology. In this paper, we mainly study the integrability of nonlinear equations with variable coefficients, the periodic wave solutions and integrability of extended KP equations, and the Hamiltonian structure of the family of evolution equations in the 21 ~ (1-D) dimension. In the introduction, we mainly introduce the emergence of soliton theory, its development and research status, and then introduce the topic and main work of this paper. In the second chapter, according to the relation between Bell polynomial and Hirota bilinear D operator, the bilinear derivative form and bilinear B?cklund transformation of variable coefficient equation are derived. Then, according to the relation between the bilinear B?cklund transform v=ln polynomials and the Hopf-Cole transformation v=ln 蠄, we derive the concrete expressions of the Lax pairs and conjugate Lax pairs of nonlinear equations. Finally, we derive the infinite conservation law of the equation. In chapter 3, by using the theory of periodic wave solution of Riemann-theta function, we obtain the periodic wave solution of extended KP equation, and make an asymptotic analysis of the obtained periodic wave solution. Finally, we obtain some integrable properties of the equation. In chapter 4, a new family of integrable equations is generated by using the methods of Li algebra and Tu scheme, and then their Hamiltonian structures are obtained by reducing them. Chapter five summarizes this paper.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175.29

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