非线性波方程的精确解与分支问题研究
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摘要
非线性波方程是描述自然现象的一类重要数学模型,也是非线性数学物理特别是孤立子理论最前沿的研究课题之一。通过对非线性波方程的求解和定性分析的研究,有助于人们弄清系统在非线性作用下的运动变化规律,合理解释相关的自然现象,更加深刻地描述系统的本质特征,极大地推动相关学科如物理学、力学、应用数学以及工程技术的发展。
本文从动力系统分支理论的角度来研究非线性波方程的精确行波解、行波解的分支及其动力学行为。首先,在现有求解非线性波方程的主要方法的基础上,对非线性波方程的精确解求解方法进行了研究,利用动力系统分支理论方法改进了求解非线性波方程精确解的一种子方程法,并用于求解几类重要的非线性数学物理方程,获得了一系列新的结果。其次,以动力系统分支理论和奇异摄动理论为研究工具,研究了几类源于实际物理问题的非线性波方程的行波解的定性行为,揭示了这些非线性模型中蕴涵的丰富的动力学性质,获得了奇异同宿轨道的动力学性质,分析并解释了这些复杂行波解产生的原因,丰富和发展了李继彬教授提出的研究奇异非线性波方程的动力系统方法一三步法。
本文主要研究工作如下:
第一章是绪言,综述了非线性波方程的发展历史、研究现状、主要研究方法以及取得的成果,介绍了近年来非光滑波的发现、相应的研究方法及其最新研究进展,指出了非线性波方程与动力系统之间的联系以及运用动力系统相关理论研究非线性波方程的现状。本章最后介绍了李继彬教授提出的研究非线性波方程的“三步法”的主要理论和结果以及其它预备知识。
第二章通过改进范恩贵教授提出的求解非线性波方程的一种子方程法,研究了Sawada-Kotera方程的求解问题。该子方程法通过在复杂非线性方程与相对简单的一个子方程之间巧妙地构造一个多项式变换,把求解非线性波方程的问题转化为求解子方程。因此如何获得子方程的更多的精确解成为该方法的关键步骤。本文利用动力系统分支理论研究了一般形式的子方程,提出了改进的子方程法,并将之应用于求解Sawada-Kotera方程,获得了Sawada-Kotern方程的大量新精确解,如多峰孤立波解,多峰周期行波解等。特别地,在所获得的精确解中所含参数都与方程的系统参数无关,因此,让这些参数取不同的值,相应的解便会呈现十分丰富的动力学行为。利用这种改进的方法求解非线性波方程的优越之处在于,我们不仅可以获得一般形式的子方程的所有精确解(为节省篇幅,本文主要给出了它们的所有孤立波解和扭波解、部分的有理解和周期行波解),而且还能获悉每一个解的动力学性质及其满足的参数条件,这充分显示了利用动力系统分支理论改进的方法在研究非线性波方程精确解方面的优越性和有效性。
第三章利用“三步法”研究了一类正则长波方程即R(m,n)方程的行波解。利用时间尺度变换,把R(m,n)方程的奇异行波系统转化为一个正则动力系统,在运用经典的动力系统分支理论研究正则系统的轨道的定性行为的基础上,利用正则系统与奇异系统之间的联系以及奇异摄动理论知识获得了R(m,n)方程行波解的定性信息,解释了该方程非光滑行波解产生的原因,并证明了正则系统的奇异同宿轨道对应的解是R(m,n)方程的光滑周期行波解而不是孤立波解。
第四章研究了一类非线性耗散项和非线性色散项共存的n+1维Klein-Gordon方程,讨论了非线性耗散强度、非线性色散强度和非线性强度效应的共同作用对系统的影响,这种影响主要表现在解的动力学性质对这些非线性强度的依赖性。强调了奇异直线的存在是导致系统出现非光滑的周期尖波、孤立尖波和破缺波的根本原因,获得了各种光滑波和非光滑波存在的充分条件。奇异系统与正则系统具有不同的时间尺度,从而导致两系统某些对应轨道有着完全不同的动力学性质,比如,与正则系统的奇异同宿轨道相对应的奇异系统的轨道可能是其周期轨道也可能仍是同宿轨道,奇异系统的这两种不同的轨道对应的是原Klein-Gordon方程具有完全不同动力学性质的解:周期轨道对应着光滑的周期行波解而同宿轨道对应着光滑的孤立波解。然而如何判定奇异同宿轨道是奇异系统的周期轨道还是同宿轨道?这又依赖于非线性耗散强度和非线性色散强度。这些现象充分反映了非线性耗散强度、非线性色散强度以及非线性强度效应的共同作用对系统的本质影响,也充分展示了奇异非线性波系统的魅力。本文利用奇异摄动理论解释了正则系统与奇异系统之间对应轨道具有不同动力学行为这一奇妙现象,对其给予了严格的数学证明并给出了判定轨道性质的具体方法,丰富和发展了研究非线性波方程的动力系统方法一三步法。
第五章研究了两类变形的2+1维Boussinesq型方程(正指数Boussinesq方程和负指数Boussinesq方程)的行波解的定性行为。由于它们的行波系统都具有奇性,因此我们借助微分方程定性理论研究了对应的正则系统,获得了正则系统所有有界轨道的定性性质,进而分析了这两类方程光滑行波解和非光滑行波解产生的分支参数条件,获得了各种有界行波解存在的充分条件。特别地,对于负指数Boussinesq方程的行波系统而言,其正则系统的所有光滑轨道都对应着奇异系统的光滑轨道,正则系统的奇异同宿轨道和异宿轨道也分别对应着奇异系统的同宿轨道和异宿轨道(即负指数Boussinesq方程的光滑孤立波解和扭波解),从而得到了负指数Boussinesq方程在一定的参数条件下不可数无穷多个光滑孤立波解的存在性。对于负指数Boussinesq方程来说,奇性并没有导致非光滑行波解的出现,这说明奇异直线的存在只是使奇异系统有非光滑解存在的可能性,但并不必然导致系统出现非光滑解。也就是说,奇异行波系统不一定存在非光滑的行波解。
第六章对本文的工作进行了总结,提出了有待进一步研究的问题。
Nonlinear wave equations are important mathematical models for describing natural phenomena and are one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solutions of nonlinear wave equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.
In this dissertation, the exact travelling wave solutions, the bifurcations and dynamical behavior of travelling wave solutions of the nonlinear wave equations are investigated from the viewpoint of bifurcation theory of dynamical systems. Firstly, based on the main methods for finding exact solutions of nonlinear wave equations, the bifurcation theory and method of dynamical systems are used to improve a sub-equation method which is an effective method of finding exact solutions of nonlinear partial differential equations. By making use of the improved sub-equation method, a variety of new exact solutions to many physically significant nonlinear equations of mathematical physics are obtained. The research enriches and develops the approaches to exact solutions of differential equations. On the other hand, by using bifurcation theory of dynamical systems and singular perturbation theory, the qualitative behavior of travelling wave solutions to the several families of nonlinear wave equations from physical background are studied, and the rich dynamical properties of these nonlinear models are shown. Moreover the dynamical properties of singular homoclinic orbits are considered and first obtained, and the reason why the complex travelling wave solutions appear is successfully analyzed and explained. The results obtained develop the three-step method, an approach of dynamical systems proposed by professor Jibin Li for studying singular nonlinear wave equations.
The major works of this dissertation mainly are as follows.
In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear wave equations are summarized. The discovery, correspond- ing research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear wave equations and dynamical systems along with the study on nonlinear wave equations by using the theory of dynamical systems are presented. In the end of the chapter, some preliminary knowledge of dynamical systems and the basic mathematic theory and main results of the three-step method are introduced.
In Chapter 2, by improving a sub-equation method proposed by professor Engui Fan for finding exact solutions to nonlinear wave equations, the explicit exact solutions of the Sawada-Kotera equation are discussed. The essence of the sub-equation method is to convert the problem for finding travelling wave solutions of nonlinear wave equations to the problem for solving a sub-equation by making a wonderful polynomial transformation between the more complicated nonlinear wave equation and the simpler sub-equation. Thus how to find more exact solutions of the sub-equation becomes a key step in this method. Here by making full advantage of bifurcation theory of dynamical systems to study the sub-equation with general forms, an improved sub-equation method is presented and applied to the Sawada-Kotera equation. As a result, a series of travelling wave solutions to the Sawada-Kotera equation are obtained in a systematic way, which include multi-hump solitary wave solutions, multi-hump periodic wave solutions and so on. Moreover all the parameters in the obtained solutions are independent of the systematic parameters in the Sawada-Kotera equation, and therefore these solutions exhibit rich dynamical behavior by taking different values of these parameters. By using the improved method, not only all the solutions of the sub-equation with general forms can be gained (in the thesis, to save space, all solitary wave solutions and kink wave solutions, some rational function solutions along with some periodic travelling wave solutions are just given), but also the dynamical property and the parametric condition of each solution obtained can be known, which shows the advantage and effectiveness of the improved sub-equation method for researching exact solutions of nonlinear wave equations.
In Chapter 3, the travelling wave solutions of a family of regularized long-wave equations, i.e., R(m,n) equations, are discussed by using the three-step method. After making a transformation of time scale, the singular travelling wave system of R(m,n) equations is reduced to a regular dynamical system. And the qualitative behavior of orbits of the regular system can be obtained by using the classical bifurcation theory of dynamical systems. Hence the qualitative information of the travelling wave system of R(m,n) equations are achieved from singular perturbation theory and the relation between the singular system and the regular system. How smooth travelling wave solutions lose their smoothness and become non-smooth travelling wave solutions is explained. And the fact is proved that the travelling wave solutions of R(m,n) equations corresponding to the singular homo-clinic orbits of the regular system are not their smooth solitary wave solutions but smooth periodic travelling wave solutions.
In Chapter 4, a family of n+1-dimensional Klein-Gordon equations with nonlinear dissipative term and nonlinear dispersive term are studied. The effect on the system under the common actions of nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect is discussed, which shows that the dynamical behavior of solutions depends greatly on the nonlinear intensity. It is emphasized that the existence of singular straight line is the original reason for the appearance of non-smooth periodic cusp wave solutions, solitary cusp wave solutions and breaking wave solutions. Various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given. The singular travelling wave system and the regular travelling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. The orbits of the singular travelling wave system which correspond to the singular homoclinic orbits of the regular travelling wave system, for example, are either its periodic orbits or still homoclinic orbits. While the two distinct orbits respectively correspond to different solutions of Klein-Gordon equations with distinct dynamical behavior: the periodic orbits of singular system correspond to the periodic travelling wave solutions while the homoclinic orbits correspond to the solitary wave solutions. But how to judge the singular homoclinic orbits are periodic orbits or homoclinic orbits of singular system depends on the nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect. These phenomena show sufficiently the essential effect on the nonlinear system under the three kinds of intensity. In this chapter, by taking advantage of singular perturbation theory, the wonderful phenomena that the corresponding orbits between the singular system and the regular system have different dynamical behavior are explained and strictly proved in mathematics. And the methods for judging the orbital dynamical property are given. These results obtained here enrich and develop the three-step method.
In Chapter 5, the qualitative behavior of travelling wave solutions to two variants of 2+1 dimensional Boussinesq-type equations with positive and negative exponents respectively is studied. Since both their traveling wave systems have singularity, after applying qualitative theory of differential equations to the corresponding regular systems, the qualitative properties of all bounded orbits of the regular systems are obtained. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth traveling wave solutions to the two variants of Boussinesq-type equations are analyzed and the various sufficient conditions to guarantee the existence of the bounded traveling wave solutions are obtained. Especially, for the Boussinesq-type equations with negative exponent, all the smooth orbits of the regular system correspond to the smooth orbits of the singular system. The singular homoclinic orbits and heteroclinic orbits of the regular system also are respectively the homoclinic orbits and heteroclinic orbits of the singular system, i.e., the smooth solitary wave solutions and smooth kink (or anti-kink) wave solutions of the Boussinesq-type equations with negative exponent. Hence the existence of uncountably infinite many smooth solitary wave solutions under certain parametric conditions is obtained. For the Boussinesq-type equations with negative exponent, the singularity does not cause the appearance of non-smooth traveling wave solutions, which shows that singular line does not always result in non-smooth solutions. That is to say, singular traveling wave systems do not always exist non-smooth traveling wave solutions.
In Chapter 6, the summary of this dissertation and the prospect of future study are given.
本文从动力系统分支理论的角度来研究非线性波方程的精确行波解、行波解的分支及其动力学行为。首先,在现有求解非线性波方程的主要方法的基础上,对非线性波方程的精确解求解方法进行了研究,利用动力系统分支理论方法改进了求解非线性波方程精确解的一种子方程法,并用于求解几类重要的非线性数学物理方程,获得了一系列新的结果。其次,以动力系统分支理论和奇异摄动理论为研究工具,研究了几类源于实际物理问题的非线性波方程的行波解的定性行为,揭示了这些非线性模型中蕴涵的丰富的动力学性质,获得了奇异同宿轨道的动力学性质,分析并解释了这些复杂行波解产生的原因,丰富和发展了李继彬教授提出的研究奇异非线性波方程的动力系统方法一三步法。
本文主要研究工作如下:
第一章是绪言,综述了非线性波方程的发展历史、研究现状、主要研究方法以及取得的成果,介绍了近年来非光滑波的发现、相应的研究方法及其最新研究进展,指出了非线性波方程与动力系统之间的联系以及运用动力系统相关理论研究非线性波方程的现状。本章最后介绍了李继彬教授提出的研究非线性波方程的“三步法”的主要理论和结果以及其它预备知识。
第二章通过改进范恩贵教授提出的求解非线性波方程的一种子方程法,研究了Sawada-Kotera方程的求解问题。该子方程法通过在复杂非线性方程与相对简单的一个子方程之间巧妙地构造一个多项式变换,把求解非线性波方程的问题转化为求解子方程。因此如何获得子方程的更多的精确解成为该方法的关键步骤。本文利用动力系统分支理论研究了一般形式的子方程,提出了改进的子方程法,并将之应用于求解Sawada-Kotera方程,获得了Sawada-Kotern方程的大量新精确解,如多峰孤立波解,多峰周期行波解等。特别地,在所获得的精确解中所含参数都与方程的系统参数无关,因此,让这些参数取不同的值,相应的解便会呈现十分丰富的动力学行为。利用这种改进的方法求解非线性波方程的优越之处在于,我们不仅可以获得一般形式的子方程的所有精确解(为节省篇幅,本文主要给出了它们的所有孤立波解和扭波解、部分的有理解和周期行波解),而且还能获悉每一个解的动力学性质及其满足的参数条件,这充分显示了利用动力系统分支理论改进的方法在研究非线性波方程精确解方面的优越性和有效性。
第三章利用“三步法”研究了一类正则长波方程即R(m,n)方程的行波解。利用时间尺度变换,把R(m,n)方程的奇异行波系统转化为一个正则动力系统,在运用经典的动力系统分支理论研究正则系统的轨道的定性行为的基础上,利用正则系统与奇异系统之间的联系以及奇异摄动理论知识获得了R(m,n)方程行波解的定性信息,解释了该方程非光滑行波解产生的原因,并证明了正则系统的奇异同宿轨道对应的解是R(m,n)方程的光滑周期行波解而不是孤立波解。
第四章研究了一类非线性耗散项和非线性色散项共存的n+1维Klein-Gordon方程,讨论了非线性耗散强度、非线性色散强度和非线性强度效应的共同作用对系统的影响,这种影响主要表现在解的动力学性质对这些非线性强度的依赖性。强调了奇异直线的存在是导致系统出现非光滑的周期尖波、孤立尖波和破缺波的根本原因,获得了各种光滑波和非光滑波存在的充分条件。奇异系统与正则系统具有不同的时间尺度,从而导致两系统某些对应轨道有着完全不同的动力学性质,比如,与正则系统的奇异同宿轨道相对应的奇异系统的轨道可能是其周期轨道也可能仍是同宿轨道,奇异系统的这两种不同的轨道对应的是原Klein-Gordon方程具有完全不同动力学性质的解:周期轨道对应着光滑的周期行波解而同宿轨道对应着光滑的孤立波解。然而如何判定奇异同宿轨道是奇异系统的周期轨道还是同宿轨道?这又依赖于非线性耗散强度和非线性色散强度。这些现象充分反映了非线性耗散强度、非线性色散强度以及非线性强度效应的共同作用对系统的本质影响,也充分展示了奇异非线性波系统的魅力。本文利用奇异摄动理论解释了正则系统与奇异系统之间对应轨道具有不同动力学行为这一奇妙现象,对其给予了严格的数学证明并给出了判定轨道性质的具体方法,丰富和发展了研究非线性波方程的动力系统方法一三步法。
第五章研究了两类变形的2+1维Boussinesq型方程(正指数Boussinesq方程和负指数Boussinesq方程)的行波解的定性行为。由于它们的行波系统都具有奇性,因此我们借助微分方程定性理论研究了对应的正则系统,获得了正则系统所有有界轨道的定性性质,进而分析了这两类方程光滑行波解和非光滑行波解产生的分支参数条件,获得了各种有界行波解存在的充分条件。特别地,对于负指数Boussinesq方程的行波系统而言,其正则系统的所有光滑轨道都对应着奇异系统的光滑轨道,正则系统的奇异同宿轨道和异宿轨道也分别对应着奇异系统的同宿轨道和异宿轨道(即负指数Boussinesq方程的光滑孤立波解和扭波解),从而得到了负指数Boussinesq方程在一定的参数条件下不可数无穷多个光滑孤立波解的存在性。对于负指数Boussinesq方程来说,奇性并没有导致非光滑行波解的出现,这说明奇异直线的存在只是使奇异系统有非光滑解存在的可能性,但并不必然导致系统出现非光滑解。也就是说,奇异行波系统不一定存在非光滑的行波解。
第六章对本文的工作进行了总结,提出了有待进一步研究的问题。
Nonlinear wave equations are important mathematical models for describing natural phenomena and are one of the forefront topics in the studies of nonlinear mathematical physics, especially in the studies of soliton theory. The research on finding explicit and exact solutions of nonlinear wave equations and on analyzing the qualitative behavior of solutions of nonlinear wave equations can help us understand the motion laws of the nonlinear systems under the nonlinear interactions, explain the corresponding natural phenomena reasonably, describe the essential properties of the nonlinear systems more deeply, and promote greatly the development of engineering technology and related subjects such as physics, mechanics and applied mathematics.
In this dissertation, the exact travelling wave solutions, the bifurcations and dynamical behavior of travelling wave solutions of the nonlinear wave equations are investigated from the viewpoint of bifurcation theory of dynamical systems. Firstly, based on the main methods for finding exact solutions of nonlinear wave equations, the bifurcation theory and method of dynamical systems are used to improve a sub-equation method which is an effective method of finding exact solutions of nonlinear partial differential equations. By making use of the improved sub-equation method, a variety of new exact solutions to many physically significant nonlinear equations of mathematical physics are obtained. The research enriches and develops the approaches to exact solutions of differential equations. On the other hand, by using bifurcation theory of dynamical systems and singular perturbation theory, the qualitative behavior of travelling wave solutions to the several families of nonlinear wave equations from physical background are studied, and the rich dynamical properties of these nonlinear models are shown. Moreover the dynamical properties of singular homoclinic orbits are considered and first obtained, and the reason why the complex travelling wave solutions appear is successfully analyzed and explained. The results obtained develop the three-step method, an approach of dynamical systems proposed by professor Jibin Li for studying singular nonlinear wave equations.
The major works of this dissertation mainly are as follows.
In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear wave equations are summarized. The discovery, correspond- ing research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear wave equations and dynamical systems along with the study on nonlinear wave equations by using the theory of dynamical systems are presented. In the end of the chapter, some preliminary knowledge of dynamical systems and the basic mathematic theory and main results of the three-step method are introduced.
In Chapter 2, by improving a sub-equation method proposed by professor Engui Fan for finding exact solutions to nonlinear wave equations, the explicit exact solutions of the Sawada-Kotera equation are discussed. The essence of the sub-equation method is to convert the problem for finding travelling wave solutions of nonlinear wave equations to the problem for solving a sub-equation by making a wonderful polynomial transformation between the more complicated nonlinear wave equation and the simpler sub-equation. Thus how to find more exact solutions of the sub-equation becomes a key step in this method. Here by making full advantage of bifurcation theory of dynamical systems to study the sub-equation with general forms, an improved sub-equation method is presented and applied to the Sawada-Kotera equation. As a result, a series of travelling wave solutions to the Sawada-Kotera equation are obtained in a systematic way, which include multi-hump solitary wave solutions, multi-hump periodic wave solutions and so on. Moreover all the parameters in the obtained solutions are independent of the systematic parameters in the Sawada-Kotera equation, and therefore these solutions exhibit rich dynamical behavior by taking different values of these parameters. By using the improved method, not only all the solutions of the sub-equation with general forms can be gained (in the thesis, to save space, all solitary wave solutions and kink wave solutions, some rational function solutions along with some periodic travelling wave solutions are just given), but also the dynamical property and the parametric condition of each solution obtained can be known, which shows the advantage and effectiveness of the improved sub-equation method for researching exact solutions of nonlinear wave equations.
In Chapter 3, the travelling wave solutions of a family of regularized long-wave equations, i.e., R(m,n) equations, are discussed by using the three-step method. After making a transformation of time scale, the singular travelling wave system of R(m,n) equations is reduced to a regular dynamical system. And the qualitative behavior of orbits of the regular system can be obtained by using the classical bifurcation theory of dynamical systems. Hence the qualitative information of the travelling wave system of R(m,n) equations are achieved from singular perturbation theory and the relation between the singular system and the regular system. How smooth travelling wave solutions lose their smoothness and become non-smooth travelling wave solutions is explained. And the fact is proved that the travelling wave solutions of R(m,n) equations corresponding to the singular homo-clinic orbits of the regular system are not their smooth solitary wave solutions but smooth periodic travelling wave solutions.
In Chapter 4, a family of n+1-dimensional Klein-Gordon equations with nonlinear dissipative term and nonlinear dispersive term are studied. The effect on the system under the common actions of nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect is discussed, which shows that the dynamical behavior of solutions depends greatly on the nonlinear intensity. It is emphasized that the existence of singular straight line is the original reason for the appearance of non-smooth periodic cusp wave solutions, solitary cusp wave solutions and breaking wave solutions. Various sufficient conditions to guarantee the existence of smooth and non-smooth travelling wave solutions are given. The singular travelling wave system and the regular travelling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. The orbits of the singular travelling wave system which correspond to the singular homoclinic orbits of the regular travelling wave system, for example, are either its periodic orbits or still homoclinic orbits. While the two distinct orbits respectively correspond to different solutions of Klein-Gordon equations with distinct dynamical behavior: the periodic orbits of singular system correspond to the periodic travelling wave solutions while the homoclinic orbits correspond to the solitary wave solutions. But how to judge the singular homoclinic orbits are periodic orbits or homoclinic orbits of singular system depends on the nonlinear dissipative intensity, nonlinear dispersive intensity and nonlinear intensity effect. These phenomena show sufficiently the essential effect on the nonlinear system under the three kinds of intensity. In this chapter, by taking advantage of singular perturbation theory, the wonderful phenomena that the corresponding orbits between the singular system and the regular system have different dynamical behavior are explained and strictly proved in mathematics. And the methods for judging the orbital dynamical property are given. These results obtained here enrich and develop the three-step method.
In Chapter 5, the qualitative behavior of travelling wave solutions to two variants of 2+1 dimensional Boussinesq-type equations with positive and negative exponents respectively is studied. Since both their traveling wave systems have singularity, after applying qualitative theory of differential equations to the corresponding regular systems, the qualitative properties of all bounded orbits of the regular systems are obtained. Therefore the bifurcation parameter conditions which lead to smooth and non-smooth traveling wave solutions to the two variants of Boussinesq-type equations are analyzed and the various sufficient conditions to guarantee the existence of the bounded traveling wave solutions are obtained. Especially, for the Boussinesq-type equations with negative exponent, all the smooth orbits of the regular system correspond to the smooth orbits of the singular system. The singular homoclinic orbits and heteroclinic orbits of the regular system also are respectively the homoclinic orbits and heteroclinic orbits of the singular system, i.e., the smooth solitary wave solutions and smooth kink (or anti-kink) wave solutions of the Boussinesq-type equations with negative exponent. Hence the existence of uncountably infinite many smooth solitary wave solutions under certain parametric conditions is obtained. For the Boussinesq-type equations with negative exponent, the singularity does not cause the appearance of non-smooth traveling wave solutions, which shows that singular line does not always result in non-smooth solutions. That is to say, singular traveling wave systems do not always exist non-smooth traveling wave solutions.
In Chapter 6, the summary of this dissertation and the prospect of future study are given.
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