若干非线性微分方程的对称与守恒律及解析解的研究

发布时间：2018-04-16 21:07

本文选题：李对称分析 + 解析解　；参考：《中国矿业大学》2017年硕士论文

【摘要】：本文主要研究几类非线性微分方程的对称,守恒律与解析解.首先简单介绍了相关的研究背景和本文的主要工作.然后,将李对称方法推广到一种压力波Kudryashov-Sinelshchikov方程上,求出其无穷小对称的向量场与群不变解,并在此基础上得到了该方程的精确解析幂级数解.第三章和第四章,对广义的Korteweg-de Vries-Fischer方程进行详细的对称分类分析,分析出该方程无穷小对称的向量场.在此基础上研究其自伴随性质,最后基于伴随方程法和守恒乘子直接构造法的相关理论,系统地研究该方程的守恒律.第五章和第六章,基于Hirota双线性法,将Bell多项式与黎曼theta函数推广到(3+1)-维广义的B-type Kadomtsev-Petviashvili方程中,得到了该方程的双线性形式和解析解,包括孤子解和周期波解.并进一步研究所求出周期波解的渐近性质,证明了满足某种极限条件下,周期波解退化成孤子解.并对方程的孤子解,周期波解以及周期波解的渐近情况进行图形模拟与分析.接下来,基于可积离散化理论,研究了薛定谔型方程Eckhaus-Kundu方程的半离散和全离散,并给出相应的孤子解以及孤子解的图形模拟.最后对全文进行简单的总结和展望.
[Abstract]:In this paper, the symmetry, conservation laws and analytical solutions of some nonlinear differential equations are studied.Firstly, the research background and the main work of this paper are briefly introduced.Then, the lie symmetry method is extended to a pressure wave Kudryashov-Sinelshchikov equation, and its infinitesimal symmetric vector field and group invariant solutions are obtained, and the exact analytic power series solutions of the equation are obtained.In the third and fourth chapters, the generalized Korteweg-de Vries-Fischer equation is analyzed in detail, and the vector fields of the equation are analyzed.On this basis, the self-adjoint property of the equation is studied. Finally, the conservation law of the equation is studied systematically based on the related theories of the adjoint equation method and the direct construction method of the conservation multiplier.In the fifth and sixth chapters, based on the Hirota bilinear method, the Bell polynomial and Riemannian theta function are extended to the generalized B-type Kadomtsev-Petviashvili equation with 31- dimension. The bilinear form and analytic solution of the equation are obtained, including soliton solution and periodic wave solution.Furthermore, the asymptotic properties of periodic wave solutions are studied, and it is proved that the periodic wave solutions degenerate into soliton solutions under certain limit conditions.The asymptotic behavior of soliton solution, periodic wave solution and periodic wave solution of the equation are simulated and analyzed graphically.Then, based on the integrable discretization theory, the semi-discrete and fully discrete Schrodinger equation Eckhaus-Kundu equations are studied, and the corresponding soliton solutions and the graphical simulation of the soliton solutions are given.At last, the paper makes a brief summary and prospect.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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