非线性演化方程的精确解及其动力学行为研究
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摘要
随着科学技术的发展,在自然科学和社会科学领域中广泛存在的非线性问题,越来越引起人们的关注,而且许多非线性问题的研究最终可归结为非线性演化方程来描述,通过对非线性演化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演化方程的精确行波解、行波解的分支及其动力学行为,主要研究工作如下：
第一章是绪论,综述了非线性演化方程的发展历史、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,指出了非线性演化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演化方程的现状。最后介绍了研究非线性演化方程的动力系统方法一“三步法”的主要理论和结果以及其它预备知识。
第二章利用动力系统方法研究了n+1维双sine-Gordon(DSG)方程与双sinh-Gordon(DSHG)方程的精确行波解。首先分别在相柱面和相平面上研究了DSG方程和DSHG方程的动力学性质,得到了这两个方程在不同参数空间的所有可能的精确行波解。随后利用三个不同的变换对这两个方程作进一步探究,在某些变换后得到的行波系统具有奇性,通过时间尺度变换消除奇性,把奇异系统化为正则系统,运用经典的动力系统分支理论研究了正则系统的轨道的定性行为,再利用奇异摄动理论分析了正则系统与奇异系统的轨道之间的关系,获得了奇异系统的解的动力学性质,我们得到这样一个重要事实：DSG方程与DSHG方程在这些变换下的行波解都是光滑的。结合相平面分析,我们给出了DSG方程与DSHG方程在这些变换下的所有可能的精确行波解。最后经过逐一验证,说明在变换下求出的DSG方程与DSHG方程的解都包含在不作变换而直接求解原方程所得到的解当中。也就是说,通过这些变换求出的DSG方程和DSHG方程的行波解只是形式上发生了变化,变换从本质上并没有改变原方程的动力学性质。充分说明了动力系统方法是研究非线性演化方程的行波解的有效方法,通过研究系统的解的动力学性质,所得到的行波解全面而细致,这是其他方法不可替代的。
第三章研究了广义Calogero-Degasperis-Fokas (CDF)方程的动力学性质与精确行波解。由于原行波系统具有奇性,通过时间尺度变换消除奇性并化为正则系统后,不同的时间尺度导致两系统某些对应轨道有着不同的动力学性质,利用奇异摄动理论分析了正则系统与奇异系统的轨道之间的关系,证明了正则系统的奇异同宿轨道在不同参数条件下分别对应着奇异系统的周期轨道和同宿轨道,而正则系统的异宿轨道在不同参数条件下对应着奇异系统的同宿轨道和异宿轨道,说明了奇直线的存在只是使得系统“有可能”存在非光滑解,并非必然导致系统出现非光滑解,并解释了破缺波产生的原因,得到了广义CDF方程在不同参数空间所有可能的精确行波解的显式表达式,这些解既包含光滑的孤立波、扭波、反扭波和周期波,也包含非光滑双边破缺的峰(谷)型破缺波与单边破缺的破缺扭波和反扭波。说明了奇直线的存在使得非线性演化方程的行波解呈现非常复杂的动力学行为,而动力系统方法恰是研究这些复杂而有趣的现象的有效工具。
第四章研究了广义非线性导数Schrodinger方程和高阶色散非线性Schrodinger方程的精确行波解,根据这两类方程的实际物理背景,通过适当的行波变换,把对这两类方程的行波解研究统一为对同一个Hamilton系统的研究,通过对该Hamilton系统的动力学行为完整而细致的讨论,得到了这两类Schrodinger型方程在不同参数条件下所有可能的包络孤立波解、包络扭波解和周期波解,所得到的结果比其他文献中的更为完整。
第五章研究了非线性色散Schrodinger方程,即NLS(m,n)方程的精确解。通过适当的变换,把对复非线性演化方程的研究转化为对平面可积系统的研究,运用经典的平面动力系统的分支理论方法系统地研究了NLS(m,n)方程的解的动力学行为,解释了非光滑的周期尖斑图解和破缺斑图解出现的原因,获得了各种光滑解和非光滑解存在的充分条件,得到了NLS(m,n)方程的一些精确解的显式和隐式表达式,这些解既包含光滑的包络孤波斑图解、包络扭波斑图解和周期斑图解,也包含非光滑的周期尖斑图解。
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。
With the development of science and technology, there are many nonlinear problems in natural and social areas, which arouses much concern. These problems are usually characterized by nonlinear evolution partial differential equations. The research on find-ing explicit and exact solutions of nonlinear evolution 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 corre-sponding 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 dynam-ical behavior of travelling wave solutions of the nonlinear evolution equations are inves-tigated from the viewpoint of bifurcation theory of dynamical systems. The major works of this dissertation mainly are as follows.
In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear evolution equations are summarized. The discovery, corre-sponding research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear evolution equations and dynamical systems along with the study on nonlinear evolution 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, the exact travelling wave solutions of the n+1 dimensional double sine-Gordon(DSG) equation and double sinh-Gordon(DSHG) equation are studied by using the dynamical system approach. First, we study the dynamical behavior of DSG equation and DSHG equation on phase cylinder and phase plane respectively and obtain all pos-sible explicit exact travelling wave solutions of the two equations in different parametric regions. Second, we investigate the two equations by using three different transforma-tions. Under some transformations, the resulting travelling wave systems are singular. After making a transformation of time scale, the singular systems are reduced to the regu-lar 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 dynamical behav-ior of the solutions of the singular systems are achieved from singular perturbation theory and the relation between the singular system and the regular system. An important fact is obtained:under these transformations the travelling wave solutions of DSG and DSHG equations are smooth. By the analysis of phase plane we give all possible explicit exact travelling wave solutions of the DSG and DSHG equations under these transformations. Finally, after verifying one by one, we illustrate that the solutions of DSG and DSHG equations which we obtained by making transformations are included in those obtained by considering the original equation directly. In other words, the solutions obtained by using transformations are just changed in forms and the transformations do not change the dynamical behavior of the equations essentially. We can conclude that the dynamical system approach is an effective method to solve nonlinear evolution equations. The trav-elling wave solutions obtained by using dynamical system approach are comprehensive and the approach can not be displaced by other methods.
In Chapter 3, the exact travelling wave solutions and dynamical behavior of the gen-eralized Calogero-Degasperis-Fokas (gCDF) equation are studied. After making a trans-formation of time scale, the singular travelling wave system of gCDF equation is reduced to a regular dynamical system. The singular travelling wave system and the regular trav-elling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. And the relationship of orbits between the regular system and the singular system is analyzed by using singular perturbation theory. And the fact is proved that the singular homoclinic orbit of the regular system is corresponding to smooth periodic orbit or homoclinic orbit of the singular system; the heteroclinic orbit of the regular system is corresponding to smooth homoclinic orbit or heteroclicic orbit of the singular system. It shows that singular line does not always result in non-smooth so-lutions. Furthermore, the reason of the occurrence of breaking wave is explained. Finally, we give all possible explicit exact travelling wave solutions of gCDF equation, which in-clude not only smooth solitary waves, kink waves, anti-kink waves and periodic waves but also breaking waves with peak type or valley type (breaking in two sides), breaking kink waves and breaking anti-kink waves(breaking in one side). The existence of singu-lar line makes the dynamical behavior of travelling wave solutions of nonlinear evolution equations be more complex and the dynamical system approach is the effective tool to study these complex and interesting problems.
In Chapter 4, the exact travelling wave solutions of the generalized nonlinear deriva-tive Schrodinger equation and the high order dispersive nonlinear Schrodinger equation are studied. According to the physical background and by proper travelling wave trans-formations, the research on the two equations is reduced to the research on the same Hamiltonian system. Through complete and delicate discussion of the dynamical behav-ior of the Hamiltonian system, we obtain all possible envelope solitary wave, envelope kink (anti-kink) wave and periodic wave solutions of the two equations, which are more complete than those in other papers.
In Chapter 5, the exact solutions of the nonlinearly dispersive Schrodinger equation, i.e. NLS(m,n) equation, are studied. The research on complex nonlinear evolution equa-tion is transformed to the research on a planar integrable system by proper transformation. The dynamical behavior of solutions of NLS(m,n) equation is studied systematically by using the classical theory and methods of planar dynamical systems. The reasons for ap-pearance of non-smooth periodic cusp patterns and breaking patterns are explained and the sufficient conditions to guarantee the existence of smooth and non-smooth solutions are obtained. The explicit and implicit parametric representation of some smooth enve-lope solitary patterns, envelope kink patterns, periodic patterns and non-smooth periodic cusp patterns are given.
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演化方程的精确行波解、行波解的分支及其动力学行为,主要研究工作如下：
第一章是绪论,综述了非线性演化方程的发展历史、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,指出了非线性演化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演化方程的现状。最后介绍了研究非线性演化方程的动力系统方法一“三步法”的主要理论和结果以及其它预备知识。
第二章利用动力系统方法研究了n+1维双sine-Gordon(DSG)方程与双sinh-Gordon(DSHG)方程的精确行波解。首先分别在相柱面和相平面上研究了DSG方程和DSHG方程的动力学性质,得到了这两个方程在不同参数空间的所有可能的精确行波解。随后利用三个不同的变换对这两个方程作进一步探究,在某些变换后得到的行波系统具有奇性,通过时间尺度变换消除奇性,把奇异系统化为正则系统,运用经典的动力系统分支理论研究了正则系统的轨道的定性行为,再利用奇异摄动理论分析了正则系统与奇异系统的轨道之间的关系,获得了奇异系统的解的动力学性质,我们得到这样一个重要事实：DSG方程与DSHG方程在这些变换下的行波解都是光滑的。结合相平面分析,我们给出了DSG方程与DSHG方程在这些变换下的所有可能的精确行波解。最后经过逐一验证,说明在变换下求出的DSG方程与DSHG方程的解都包含在不作变换而直接求解原方程所得到的解当中。也就是说,通过这些变换求出的DSG方程和DSHG方程的行波解只是形式上发生了变化,变换从本质上并没有改变原方程的动力学性质。充分说明了动力系统方法是研究非线性演化方程的行波解的有效方法,通过研究系统的解的动力学性质,所得到的行波解全面而细致,这是其他方法不可替代的。
第三章研究了广义Calogero-Degasperis-Fokas (CDF)方程的动力学性质与精确行波解。由于原行波系统具有奇性,通过时间尺度变换消除奇性并化为正则系统后,不同的时间尺度导致两系统某些对应轨道有着不同的动力学性质,利用奇异摄动理论分析了正则系统与奇异系统的轨道之间的关系,证明了正则系统的奇异同宿轨道在不同参数条件下分别对应着奇异系统的周期轨道和同宿轨道,而正则系统的异宿轨道在不同参数条件下对应着奇异系统的同宿轨道和异宿轨道,说明了奇直线的存在只是使得系统“有可能”存在非光滑解,并非必然导致系统出现非光滑解,并解释了破缺波产生的原因,得到了广义CDF方程在不同参数空间所有可能的精确行波解的显式表达式,这些解既包含光滑的孤立波、扭波、反扭波和周期波,也包含非光滑双边破缺的峰(谷)型破缺波与单边破缺的破缺扭波和反扭波。说明了奇直线的存在使得非线性演化方程的行波解呈现非常复杂的动力学行为,而动力系统方法恰是研究这些复杂而有趣的现象的有效工具。
第四章研究了广义非线性导数Schrodinger方程和高阶色散非线性Schrodinger方程的精确行波解,根据这两类方程的实际物理背景,通过适当的行波变换,把对这两类方程的行波解研究统一为对同一个Hamilton系统的研究,通过对该Hamilton系统的动力学行为完整而细致的讨论,得到了这两类Schrodinger型方程在不同参数条件下所有可能的包络孤立波解、包络扭波解和周期波解,所得到的结果比其他文献中的更为完整。
第五章研究了非线性色散Schrodinger方程,即NLS(m,n)方程的精确解。通过适当的变换,把对复非线性演化方程的研究转化为对平面可积系统的研究,运用经典的平面动力系统的分支理论方法系统地研究了NLS(m,n)方程的解的动力学行为,解释了非光滑的周期尖斑图解和破缺斑图解出现的原因,获得了各种光滑解和非光滑解存在的充分条件,得到了NLS(m,n)方程的一些精确解的显式和隐式表达式,这些解既包含光滑的包络孤波斑图解、包络扭波斑图解和周期斑图解,也包含非光滑的周期尖斑图解。
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。
With the development of science and technology, there are many nonlinear problems in natural and social areas, which arouses much concern. These problems are usually characterized by nonlinear evolution partial differential equations. The research on find-ing explicit and exact solutions of nonlinear evolution 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 corre-sponding 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 dynam-ical behavior of travelling wave solutions of the nonlinear evolution equations are inves-tigated from the viewpoint of bifurcation theory of dynamical systems. The major works of this dissertation mainly are as follows.
In Chapter 1, the historical background, research developments, main methods and achievements of nonlinear evolution equations are summarized. The discovery, corre-sponding research approaches and recent advance of non-smooth waves are introduced. The relation between nonlinear evolution equations and dynamical systems along with the study on nonlinear evolution 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, the exact travelling wave solutions of the n+1 dimensional double sine-Gordon(DSG) equation and double sinh-Gordon(DSHG) equation are studied by using the dynamical system approach. First, we study the dynamical behavior of DSG equation and DSHG equation on phase cylinder and phase plane respectively and obtain all pos-sible explicit exact travelling wave solutions of the two equations in different parametric regions. Second, we investigate the two equations by using three different transforma-tions. Under some transformations, the resulting travelling wave systems are singular. After making a transformation of time scale, the singular systems are reduced to the regu-lar 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 dynamical behav-ior of the solutions of the singular systems are achieved from singular perturbation theory and the relation between the singular system and the regular system. An important fact is obtained:under these transformations the travelling wave solutions of DSG and DSHG equations are smooth. By the analysis of phase plane we give all possible explicit exact travelling wave solutions of the DSG and DSHG equations under these transformations. Finally, after verifying one by one, we illustrate that the solutions of DSG and DSHG equations which we obtained by making transformations are included in those obtained by considering the original equation directly. In other words, the solutions obtained by using transformations are just changed in forms and the transformations do not change the dynamical behavior of the equations essentially. We can conclude that the dynamical system approach is an effective method to solve nonlinear evolution equations. The trav-elling wave solutions obtained by using dynamical system approach are comprehensive and the approach can not be displaced by other methods.
In Chapter 3, the exact travelling wave solutions and dynamical behavior of the gen-eralized Calogero-Degasperis-Fokas (gCDF) equation are studied. After making a trans-formation of time scale, the singular travelling wave system of gCDF equation is reduced to a regular dynamical system. The singular travelling wave system and the regular trav-elling wave system have distinct time scales which cause their some corresponding orbits to have distinct dynamical properties. And the relationship of orbits between the regular system and the singular system is analyzed by using singular perturbation theory. And the fact is proved that the singular homoclinic orbit of the regular system is corresponding to smooth periodic orbit or homoclinic orbit of the singular system; the heteroclinic orbit of the regular system is corresponding to smooth homoclinic orbit or heteroclicic orbit of the singular system. It shows that singular line does not always result in non-smooth so-lutions. Furthermore, the reason of the occurrence of breaking wave is explained. Finally, we give all possible explicit exact travelling wave solutions of gCDF equation, which in-clude not only smooth solitary waves, kink waves, anti-kink waves and periodic waves but also breaking waves with peak type or valley type (breaking in two sides), breaking kink waves and breaking anti-kink waves(breaking in one side). The existence of singu-lar line makes the dynamical behavior of travelling wave solutions of nonlinear evolution equations be more complex and the dynamical system approach is the effective tool to study these complex and interesting problems.
In Chapter 4, the exact travelling wave solutions of the generalized nonlinear deriva-tive Schrodinger equation and the high order dispersive nonlinear Schrodinger equation are studied. According to the physical background and by proper travelling wave trans-formations, the research on the two equations is reduced to the research on the same Hamiltonian system. Through complete and delicate discussion of the dynamical behav-ior of the Hamiltonian system, we obtain all possible envelope solitary wave, envelope kink (anti-kink) wave and periodic wave solutions of the two equations, which are more complete than those in other papers.
In Chapter 5, the exact solutions of the nonlinearly dispersive Schrodinger equation, i.e. NLS(m,n) equation, are studied. The research on complex nonlinear evolution equa-tion is transformed to the research on a planar integrable system by proper transformation. The dynamical behavior of solutions of NLS(m,n) equation is studied systematically by using the classical theory and methods of planar dynamical systems. The reasons for ap-pearance of non-smooth periodic cusp patterns and breaking patterns are explained and the sufficient conditions to guarantee the existence of smooth and non-smooth solutions are obtained. The explicit and implicit parametric representation of some smooth enve-lope solitary patterns, envelope kink patterns, periodic patterns and non-smooth periodic cusp patterns are given.
In Chapter 6, the summary of this dissertation and the prospect of future study are given.
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