几类非线性方程的行波解研究
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摘要
分数阶、整数阶非线性偏微分方程是描述自然现象的最基本原理的重要数学模型之一。其行波解的求解和定性行为研究,将有助于掌握系统运动状态的变化规律。
本文以源于物理问题的几类非线性分数阶、整数阶偏微分方程为研究对象,得到了这些非线性数学物理方程的一些行波解的参数表示以及定性行为,揭示了这些非线性模型蕴涵的丰富的动力学性质。针对几类分数阶偏微分方程,提出了基于时空尺度不变行波解的齐次原理,并用于具体方程的求解；对整数阶方程而言,针对具有正则行波系统的一类偏微分(Konopelchenko-Dubrovsky)方程,用平面动力系统分支理论得到了有界行波解的参数表示与分支行为；针对具有奇异行波系统的一类给定的杆方程,用李继彬教授提出的基于平面动力系统分支理论的“三步法”,得到了一定参数条件下的扭波解、周期波解和一些无界行波解的表不。
全文分为五章：
第一章是绪论和预备知识。本章主要分为两个部分,第一部分介绍分数阶微分方程的相关背景知识。综述了分数阶微积分的发展历史、基本理论、在非线性科学中的应用举例以及分数阶偏微分方程求解的研究现状。第二部分介绍了基于正则系统和奇异系统的整数阶非线性偏微分方程行波解研究的动力系统方法。
第二章是几类分数阶非线性方程的时空尺度不变行波解研究。本章首先基于求具有时空尺度不变性的行波解而提出了一个基本原理,即齐次原理。再把该原理应用于推广的分数阶Benjamin-Ono方程和等离子体中推广的分数阶Zakharov-Kuznetsov方程,得到了一定参数条件下其由幂函数表示的具有时空尺度不变性的行波解。
第三章利用平面动力系统分支理论研究了Konopelchenko-Dubrovsky方程的有界光滑行波解：孤立波解、扭波(反扭波)解、周期波解。得到了这些解存在的参数条件及其12个显式表示。
第四章研究了计入横向惯性效应后非线性弹性杆的纵波运动方程的行波解,用基于平面动力系统分支理论的“三步法”得了原系统在一定参数条件下的扭波解、周期波解和一些无界行波解的表达式52个。从讨论中发现,随着杆材料的非线性增强,杆的纵向波动的动力学行为就越复杂。
第五章对本文的工作进行了总结,并对今后的研究方向作了展望。
A fractional or integer order nonlinear partial differential equation is one of the math-ematical models which are used to describe the basic principles of nature phenomena. For the models, the research of finding the representations and investigating the dynamics of their traveling wave solutions is useful to obtain the law of behavior for the system's motions.
In this thesis, taking some nonlinear fractional or integer order equations which come from physical models as objects of research, parameter representations and qualitative dy-namical behavior of some traveling wave solutions of these equations are obtained, which reveal the rich dynamics contained by these nonlinear models. To find the traveling wave solutions expressed by power functions for some fractional partial differential equations, a homogenous principle is obtained and used to get the solutions for some given equa-tions. For some integer order partial differential (Konopelchenko-Dubrovsky) equations with regular traveling systems, the bifurcation theory method of planar dynamical sys-tems is employed to find the bounded smooth traveling wave solutions. For the given rod integer order equations with singular traveling systems, the "three-step method " based on the bifurcation theory method of planar dynamical systems is used to investigate kink wave solutions, periodic wave solutions and unbounded solutions under some parameter conditions.
There are five chapters in this thesis.
In Chapter1, the research background and main methods of investigating traveling wave solutions for nonlinear equations are summarized. There are two sections in this chapter. In Section1, for fractional order nonlinear partial differential equations, the his-torical background, basic theory, the examples for applications and present situation of solving these equations are summarized. In Section2, for integer order nonlinear par-tial differential equations with regular or singular traveling wave systems, the dynamical system method for finding traveling wave solutions is introduced.
In Chapter2, the space-time scaling invariant traveling wave solutions of some frac- tional equations are investigated. Firstly, for finding the solutions, a basic principle, i.e. homogenous principle is obtained. Secondly, the principle is used, for generalized frac-tional Benjamin-Ono equations and generalized fractional Zakharov-Kuznetsov equations in plasma, to obtain the solutions which are expressed by power functions under some pa-rameter conditions.
In Chapter3, for the (2+1) dimensional Konopelchenko-Dubrovsky equation, the bifurcation theory method of planar dynamical systems is employed to find the bounded smooth traveling wave solutions:solitary wave solutions, kink (anti-kink) wave solutions, periodic wave solutions. The parameter conditions which make the solutions exist and the exact explicit parametric representations of12solutions are obtained.
In Chapter4, for the nonlinear wave equation of longitudinal oscillation in a nonlin-ear elastic rod with lateral inertia, the "three-step method " from the view of the bifur-cation theory method of planar dynamical systems is used to investigate kink wave solu-tions, periodic wave solutions and unbounded solutions under some parameter conditions. Parametric representations of52solutions are obtained. From the qualitative behavior of the solutions, it is that the dynamical behavior of longitudinal oscillation, for the given rod, becomes more and more complex along with the enhancement of nonlinearity for the material.
In Chapter5, the summary of this thesis and the prospect of future research are given.
本文以源于物理问题的几类非线性分数阶、整数阶偏微分方程为研究对象,得到了这些非线性数学物理方程的一些行波解的参数表示以及定性行为,揭示了这些非线性模型蕴涵的丰富的动力学性质。针对几类分数阶偏微分方程,提出了基于时空尺度不变行波解的齐次原理,并用于具体方程的求解；对整数阶方程而言,针对具有正则行波系统的一类偏微分(Konopelchenko-Dubrovsky)方程,用平面动力系统分支理论得到了有界行波解的参数表示与分支行为；针对具有奇异行波系统的一类给定的杆方程,用李继彬教授提出的基于平面动力系统分支理论的“三步法”,得到了一定参数条件下的扭波解、周期波解和一些无界行波解的表不。
全文分为五章：
第一章是绪论和预备知识。本章主要分为两个部分,第一部分介绍分数阶微分方程的相关背景知识。综述了分数阶微积分的发展历史、基本理论、在非线性科学中的应用举例以及分数阶偏微分方程求解的研究现状。第二部分介绍了基于正则系统和奇异系统的整数阶非线性偏微分方程行波解研究的动力系统方法。
第二章是几类分数阶非线性方程的时空尺度不变行波解研究。本章首先基于求具有时空尺度不变性的行波解而提出了一个基本原理,即齐次原理。再把该原理应用于推广的分数阶Benjamin-Ono方程和等离子体中推广的分数阶Zakharov-Kuznetsov方程,得到了一定参数条件下其由幂函数表示的具有时空尺度不变性的行波解。
第三章利用平面动力系统分支理论研究了Konopelchenko-Dubrovsky方程的有界光滑行波解：孤立波解、扭波(反扭波)解、周期波解。得到了这些解存在的参数条件及其12个显式表示。
第四章研究了计入横向惯性效应后非线性弹性杆的纵波运动方程的行波解,用基于平面动力系统分支理论的“三步法”得了原系统在一定参数条件下的扭波解、周期波解和一些无界行波解的表达式52个。从讨论中发现,随着杆材料的非线性增强,杆的纵向波动的动力学行为就越复杂。
第五章对本文的工作进行了总结,并对今后的研究方向作了展望。
A fractional or integer order nonlinear partial differential equation is one of the math-ematical models which are used to describe the basic principles of nature phenomena. For the models, the research of finding the representations and investigating the dynamics of their traveling wave solutions is useful to obtain the law of behavior for the system's motions.
In this thesis, taking some nonlinear fractional or integer order equations which come from physical models as objects of research, parameter representations and qualitative dy-namical behavior of some traveling wave solutions of these equations are obtained, which reveal the rich dynamics contained by these nonlinear models. To find the traveling wave solutions expressed by power functions for some fractional partial differential equations, a homogenous principle is obtained and used to get the solutions for some given equa-tions. For some integer order partial differential (Konopelchenko-Dubrovsky) equations with regular traveling systems, the bifurcation theory method of planar dynamical sys-tems is employed to find the bounded smooth traveling wave solutions. For the given rod integer order equations with singular traveling systems, the "three-step method " based on the bifurcation theory method of planar dynamical systems is used to investigate kink wave solutions, periodic wave solutions and unbounded solutions under some parameter conditions.
There are five chapters in this thesis.
In Chapter1, the research background and main methods of investigating traveling wave solutions for nonlinear equations are summarized. There are two sections in this chapter. In Section1, for fractional order nonlinear partial differential equations, the his-torical background, basic theory, the examples for applications and present situation of solving these equations are summarized. In Section2, for integer order nonlinear par-tial differential equations with regular or singular traveling wave systems, the dynamical system method for finding traveling wave solutions is introduced.
In Chapter2, the space-time scaling invariant traveling wave solutions of some frac- tional equations are investigated. Firstly, for finding the solutions, a basic principle, i.e. homogenous principle is obtained. Secondly, the principle is used, for generalized frac-tional Benjamin-Ono equations and generalized fractional Zakharov-Kuznetsov equations in plasma, to obtain the solutions which are expressed by power functions under some pa-rameter conditions.
In Chapter3, for the (2+1) dimensional Konopelchenko-Dubrovsky equation, the bifurcation theory method of planar dynamical systems is employed to find the bounded smooth traveling wave solutions:solitary wave solutions, kink (anti-kink) wave solutions, periodic wave solutions. The parameter conditions which make the solutions exist and the exact explicit parametric representations of12solutions are obtained.
In Chapter4, for the nonlinear wave equation of longitudinal oscillation in a nonlin-ear elastic rod with lateral inertia, the "three-step method " from the view of the bifur-cation theory method of planar dynamical systems is used to investigate kink wave solu-tions, periodic wave solutions and unbounded solutions under some parameter conditions. Parametric representations of52solutions are obtained. From the qualitative behavior of the solutions, it is that the dynamical behavior of longitudinal oscillation, for the given rod, becomes more and more complex along with the enhancement of nonlinearity for the material.
In Chapter5, the summary of this thesis and the prospect of future research are given.
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