【作者】 张程；

【导师】 李叶舟；

【作者基本信息】 北京邮电大学 ， 系统科学， 2023， 硕士

【摘要】 非线性偏微分方程(NLPDE)可以用于模拟复杂的非线性现象,解决我们在非线性科学的不同领域中遇到的一些现实问题.目前NLPDE已成为一个非常热门的领域.本文研究了两类非线性偏微分方程(组)的解析解,并通过Mathematica、Matlab等计算机软件,画出这些解的图像,另外,对其动力学性质和物理意义做了具体的分析和讨论.文章的整体结构安排如下:第一章简单回顾非线性偏微分方程的研究背景,介绍本文中主要采用的方法和研究内容.第二章主要研究时空分数阶Boussinesq方程组,它是一类非常重要的耦合系统,通常用于模拟非线性浅水表面波现象.本章采用修正的辅助方程法求得它的新的精确解,包括(暗)周期解、三角函数解.另外,用平面系统分岔理论的知识,我们将方程组转化为平面系统的形式,根据系统的分岔条件推导出其相图,从中观察得到了与之前所得结果形式不同的精确解,如Jacobian椭圆函数解等.第三章主要研究(3+1)维拓展量子Zakharov-Kuznetsov方程,用(G’/G)-展开法和Sech-Tanh展开法得到了该方程的广义三角函数解等新的精确解.最后用Adomain分解方法得到了该方程的数值解,并验证其精确性.第四章从研究对象、思想方法、所获成果等角度总结本文与已有研究的不同之处,并指出本文的不足以及潜在的后续研究方向.更多还原

【Abstract】 Nonlinear partial differential equation(NLPDE)can simulate complex nonlinear phenomenons and solve practical problems we may encounter in different fields of nonlinear science.Currently,NLPDE has been a significantly hot field.This dissertation studies various solitary wave solutions of two kinds of nonlinear partial differential equation(s),and applies computer softwares,including Mathematica,Matlab to draw the images of these solutions.In addition,the dynamical properties and physical significance of these solutions are analyzed and discussed in details.The overall structures of the paper are arranged as follows:In Chapter 1,the research background is reviewed briefly,and the main methods used in the dissertation are introduced.In Chapter 2,we mainly study the space-time fractional Boussinesq couple system,which is a very important couple system,usually used to simulate nonlinear shallow water surface waves.We get the new exact solutions of the equations by modified auxiliary equation method,including(duck)periodical solutions and trigonometric solutions.Additionally,we transform the equations into the form of a planar system,determine all the bifurcation conditions of the system,and deduce the phase portraits of it,from which we get different new exact solutions of fractional Boussinesq equations,such as Jacobian elliptic function solutions.In Chapter 3,we mainly study the(3+1)-dimensional extended quantum Zakharov-Kuznetsov equation.Firstly,the generalized trigonometric function solutions and new traveling wave solutions of this equation are solved by(G’/G)-expansion method and Sech-Tanh expansion method.Then,the Adomain decomposition method is used to verify the accuracy of the two methods.In Chapter 4,we summarize the differences between this research and the existing ones from the perspectives of objects,methods and achievements.The inadequacies of the study are pointed out and the potential directions in future are analyzed.更多还原