非线性波动方程分岔中的若干问题分析
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摘要
非线性动力学是非线性科学的一个重要分支,非线性波动方程的精确求解及其解法研究作为非线性科学中的前沿研究课题和热点问题之一,极具挑战性。由于非线性波动方程的复杂性,求解它并无统一的方法,不过针对非线性波动方程的孤立波解,人们在研究过程中,已经发展了一些有效的研究方法,比如各种求精确解的方法、数值模拟的方法及实验研究的方法,但这些方法大都针对所研究方程的特定类型的解,而无法了解非线性方程解的全局渐近行为。本论文研究几类重要的非线性波动方程的行波解,用动力系统的分岔理论对其进行定性分析,研究系统所有可能的有界行波解,分析系统参数及奇异线对系统解的结构的影响,给出各种有界解的存在条件及解的表达式,讨论各种行波解之间的演化过程及相互作用模式,探讨其动力学行为,主要内容如下:
论文在第一章回顾了非线性波动方程研究的历史背景和研究方法。第二章讨论非线性波动方程的分岔行为,以广义KdV方程、Camassa-Holm方程和耦合Bousinesq方程为例,结合相平面分析系统的所有行波解的情况,分析参数的变化对系统解的结构的影响,讨论其在转迁集上的各种分岔模式,给出不同性质的波解及其存在条件。
论文的第三章考虑奇异性对非线性波动方程解的结构的影响,针对相空间上出现奇异线的广义KdV方程,详细讨论了奇异线的存在对解的结构的影响,分析系统中非光滑行波解产生的原因。
在对系统进行定性分析、了解了系统所有可能解的大致形态的基础上,论文第四章给出了WBK方程的精确有界解,应用动力系统理论,利用连接平衡点的闭轨线的特点结合轨线与行波之间的对应关系来研究非线性波方程行波解的有界性及精确的显式表达式。另外,在同一个参数区域中,当不同类型的波解相互共存时,分析这些解相互之间的演化机制。
然后,论文考虑不同波解之间的相互作用,目前对非线性波动方程的理论研究大都集中在单模态解上,论文第五章提出了一个新的方法,设想由方程的单模态解的非线性叠加,给出非线性波动方程的复合模态解。这些单模态解可以具有不同的性质,可以具有不同的波速,也可以是不同形式的波解,论文将作理论推导并借助Maple计算工具在具体的几个非线性方程中找到了这种复合模态解。
目前对非线性波动方程的研究一般都仅限于静态波解,即所考虑的波解的波速、振幅、波宽都是不变的,本论文考虑动态的波解,探讨非线性波动方程的动力学行为,通过能量积分式和选择适当的示性函数,将复数形式的Ginzburg-Landau方程化成为三阶常微分方程,数值模拟波解的动态行为。
本论文丰富和发展了非线性波动方程解法研究的内容,得到了许多新的结果,论文最后对所做的研究工作进行了总结,并对今后的研究方向作了展望。
Nonlinear dynamics is one of the important branches in nonlinear science, whereas as a leading subject and hot interest in nonlinear science, study on the method for finding solutions of nonlinear partial differential equations has become more and more challenging. Because of the complicity in nonlinear evolution equations, there has no systemic and uniform method for all NLEEs. Many effective methods, such as many kinds of methods for exact solutions, numerical simulations and experimental ways, have been found, but we can only understand the solutions partly and not make sure of overall situations by all these methods. In this paper, we consider all possible bounded traveling wave solutions of some important nonlinear evolution equations. Bifurcation theories are used to analyse the parameter conditions for the existence of solitary wave solutions. The basic content of this paper are given as following:
Chapter one is devoted to reviewing the history and development of the NLEEs.
Bifurcations of KdV equation, Camassa-Holm equation and coupled Bousinesq equations are researched respectively in chapter two. Transition boundaries and phase portraits in different regions are given, based on which we can obtain all possible traveling wave solutions as well as the parameter conditions.
In chapter three, we think about non-analytical (non-smooth) wave solutions. To a general KdV equation with singular curves on phase plane, we explain the reason why these non-smooth traveling wave solutions arise.
Based on the bifurcation analysis, all possible traveling wave solutions have been known qualitatively. Then we find out the exact bounded wave solutions for WBK equation in chapter four. According to the theory of dynamical system, we investigate the explicit exact traveling wave solutions of nonlinear wave equations by using the characters of the closed trajectory connecting equilibrium points and the relations between obits and traveling waves. When parameters are taken in the same region, there may exist different types of solutions. Bifurcation mechanism between these solutions are revealed.
Then we focus on interactions between different waves. A new method is proposed in chapter five. By nonlinear superposition of different single-mode waves, new types of multiple-mode waves can be derived. Several cases for the two-mode waves are obtained upon using the computer language Maple.
Steady wave solutions with constant velocity, amplitude and width have fully been understood. Now we investigate the dynamical behavior of the cubic-quintic complex Ginzburg-Landau equation in this paper. Based on the assumption of a special trial function, a three-dimensional vector field has been derived from the infinite-dimensional dissipative system. Numerical simulations are used to reveal the complexity of the vector field.
By making use of the approaches proposed by us, a variety of exact solutions to many significant nonlinear evolution equations are easily presented. Finally, the summary of this dissertation and the prospect of study on the nonlinear evolution equations are given.
论文在第一章回顾了非线性波动方程研究的历史背景和研究方法。第二章讨论非线性波动方程的分岔行为,以广义KdV方程、Camassa-Holm方程和耦合Bousinesq方程为例,结合相平面分析系统的所有行波解的情况,分析参数的变化对系统解的结构的影响,讨论其在转迁集上的各种分岔模式,给出不同性质的波解及其存在条件。
论文的第三章考虑奇异性对非线性波动方程解的结构的影响,针对相空间上出现奇异线的广义KdV方程,详细讨论了奇异线的存在对解的结构的影响,分析系统中非光滑行波解产生的原因。
在对系统进行定性分析、了解了系统所有可能解的大致形态的基础上,论文第四章给出了WBK方程的精确有界解,应用动力系统理论,利用连接平衡点的闭轨线的特点结合轨线与行波之间的对应关系来研究非线性波方程行波解的有界性及精确的显式表达式。另外,在同一个参数区域中,当不同类型的波解相互共存时,分析这些解相互之间的演化机制。
然后,论文考虑不同波解之间的相互作用,目前对非线性波动方程的理论研究大都集中在单模态解上,论文第五章提出了一个新的方法,设想由方程的单模态解的非线性叠加,给出非线性波动方程的复合模态解。这些单模态解可以具有不同的性质,可以具有不同的波速,也可以是不同形式的波解,论文将作理论推导并借助Maple计算工具在具体的几个非线性方程中找到了这种复合模态解。
目前对非线性波动方程的研究一般都仅限于静态波解,即所考虑的波解的波速、振幅、波宽都是不变的,本论文考虑动态的波解,探讨非线性波动方程的动力学行为,通过能量积分式和选择适当的示性函数,将复数形式的Ginzburg-Landau方程化成为三阶常微分方程,数值模拟波解的动态行为。
本论文丰富和发展了非线性波动方程解法研究的内容,得到了许多新的结果,论文最后对所做的研究工作进行了总结,并对今后的研究方向作了展望。
Nonlinear dynamics is one of the important branches in nonlinear science, whereas as a leading subject and hot interest in nonlinear science, study on the method for finding solutions of nonlinear partial differential equations has become more and more challenging. Because of the complicity in nonlinear evolution equations, there has no systemic and uniform method for all NLEEs. Many effective methods, such as many kinds of methods for exact solutions, numerical simulations and experimental ways, have been found, but we can only understand the solutions partly and not make sure of overall situations by all these methods. In this paper, we consider all possible bounded traveling wave solutions of some important nonlinear evolution equations. Bifurcation theories are used to analyse the parameter conditions for the existence of solitary wave solutions. The basic content of this paper are given as following:
Chapter one is devoted to reviewing the history and development of the NLEEs.
Bifurcations of KdV equation, Camassa-Holm equation and coupled Bousinesq equations are researched respectively in chapter two. Transition boundaries and phase portraits in different regions are given, based on which we can obtain all possible traveling wave solutions as well as the parameter conditions.
In chapter three, we think about non-analytical (non-smooth) wave solutions. To a general KdV equation with singular curves on phase plane, we explain the reason why these non-smooth traveling wave solutions arise.
Based on the bifurcation analysis, all possible traveling wave solutions have been known qualitatively. Then we find out the exact bounded wave solutions for WBK equation in chapter four. According to the theory of dynamical system, we investigate the explicit exact traveling wave solutions of nonlinear wave equations by using the characters of the closed trajectory connecting equilibrium points and the relations between obits and traveling waves. When parameters are taken in the same region, there may exist different types of solutions. Bifurcation mechanism between these solutions are revealed.
Then we focus on interactions between different waves. A new method is proposed in chapter five. By nonlinear superposition of different single-mode waves, new types of multiple-mode waves can be derived. Several cases for the two-mode waves are obtained upon using the computer language Maple.
Steady wave solutions with constant velocity, amplitude and width have fully been understood. Now we investigate the dynamical behavior of the cubic-quintic complex Ginzburg-Landau equation in this paper. Based on the assumption of a special trial function, a three-dimensional vector field has been derived from the infinite-dimensional dissipative system. Numerical simulations are used to reveal the complexity of the vector field.
By making use of the approaches proposed by us, a variety of exact solutions to many significant nonlinear evolution equations are easily presented. Finally, the summary of this dissertation and the prospect of study on the nonlinear evolution equations are given.
引文
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