应用拓展同宿试验法求解几类非线性发展方程

发布时间：2018-04-13 00:21

本文选题：非线性发展方程 + 精确解　；参考：《内蒙古师范大学》2017年硕士论文

【摘要】：在研究非线性自然现象时,非线性发展方程扮演着至关重要的角色,其求解方法也在不断完善中.本文主要用拓展同宿试验法来求解几类非线性发展方程,主要有五章内容构成.第一章是绪论部分,主要介绍非线性发展方程的研究意义和拓展同宿试验法的研究现状,以及离散方程的简单说明,最后介绍本文主要工作.第二章将拓展同宿试验法应用于求解常系数非线性方程.由(2+1)维耗散Zabolotskaya-Khokhlov方程的常数平衡解出发,应用拓展的同宿试验法给出该方程的新的孤立波解,同时对得到的二孤波解进行分析,得出新的力学特征.通过拓展同宿试验法,构造测试函数给出Boussinesq方程的精确解.应用同宿呼吸子极限方法,找出Boussinesq方程的呼吸子孤立波解和有理呼吸波解,并发现有理呼吸波解恰好是Boussinesq方程的怪波解.第三章应用拓展同宿试验法求解变系数非线性方程.具体求解(2+1)维变系数Zakharov-Kuznetsov方程和(2+1)维BKP方程.借助Maple符号计算系统,在(2+1)维变系数Zakharov-Kuznetsov方程和(2+1)维BKP方程双线性形式的基础上,引入新的测试函数推广拓展同宿试验法而给出(2+1)维变系数Zakharov-Kuznetsov方程和(2+1)维BKP方程的几种精确解,其中包含类周期孤波解、类孤波解和类周期波解.第四章应用拓展同宿试验法求解非线性差分微分方程和分数阶非线性偏微分方程.第二节取新的测试函数并借助Maple软件给出离散KdV方程和Toda链方程的精确解.第三节通过变换将分数阶偏微分方程化成整数阶偏微分方程,再运用拓展同宿试验法求得分数阶KdV方程的周期孤波解,二孤波解.第五章是结论与展望部分.
[Abstract]:Nonlinear evolution equations play an important role in the study of nonlinear natural phenomena, and their solving methods are being improved.In this paper, the extended homoclinic test method is used to solve several kinds of nonlinear evolution equations, which consists of five chapters.The first chapter is the introduction, which mainly introduces the research significance of nonlinear evolution equation and the present situation of the extended homoclinic test method, as well as the simple explanation of the discrete equation. Finally, the main work of this paper is introduced.In the second chapter, the extended homoclinic test method is applied to solve the nonlinear equations with constant coefficients.Starting from the constant equilibrium solution of the dissipative Zabolotskaya-Khokhlov equation, a new solitary wave solution of the equation is obtained by using the extended homoclinic test method. The obtained second solitary wave solution is analyzed and the new mechanical characteristics are obtained.By extending the homoclinic test method, the exact solution of the Boussinesq equation is obtained by constructing the test function.By using homoclinic respiratory limit method, the solitary wave solution and rational respiratory wave solution of Boussinesq equation are found, and it is found that the rational respiratory wave solution is the odd wave solution of Boussinesq equation.In chapter 3, the extended homoclinic test method is used to solve the nonlinear equations with variable coefficients.The Zakharov-Kuznetsov equation with variable coefficients and the BKP equation with 21) dimension are solved.With the aid of Maple symbolic computing system, based on the bilinear form of the Zakharov-Kuznetsov equation with variable coefficients and the BKP equation of 21) dimension,This paper introduces a new test function to extend the homoclinic test method and gives some exact solutions of the Zakharov-Kuznetsov equation with variable coefficients and the BKP equation of 21) dimension, including the quasi-periodic solitary wave solution, the similar solitary wave solution and the quasi-periodic wave solution.In chapter 4, the extended homoclinic test method is used to solve nonlinear difference differential equations and fractional nonlinear partial differential equations.In the second section, the exact solutions of the discrete KdV equation and the Toda chain equation are obtained by using the new test function and the Maple software.In the third section, fractional partial differential equations are transformed into integer partial differential equations, and then the periodic solitary wave solutions and the second solitary wave solutions of fractional KdV equations are obtained by extended homoclinic test.The fifth chapter is the conclusion and prospect part.
【学位授予单位】：内蒙古师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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