【作者】 余丽琴；

【导师】 田立新；

【作者基本信息】 江苏大学 ， 系统工程， 2012， 博士

【摘要】 非线性波动方程是非线性科学研究的一个重要分支,其求解问题一直是非线性科学研究中的前沿和热点。非线性波动方程精确解的研究不仅有助于理解孤立子理论的本质属性和代数结构,而且对相应自然现象的合理解释及实际应用将起到重要的作用。由于非线性波动方程的复杂性,致使方程求解目前并无统一的、系统的方法,尽管现在己经发展了一些有效的研究方法,但这些方法大多是解决方程特定类型的解,对非线性波动方程解的全局渐近行为无法做到全面了解,并且给出的解大多不能明确其是否有界。本论文利用首次积分法和动力系统的分岔方法研究了非线性波动方程的有界行波解,包括Degasperis-Procesi方程、带有色散项的Degasperis-Procesi方程以及双组份Degasperis-Procesi方程。利用动力系统的分岔方法在解决非线性波动方程求解问题时,不仅可以区分出系统轨道的有界性以及对应行波解的有界性,而且还可同时给出这些行波解产生的参数条件以及解的形状。本文的研究丰富和发展了非线性波动方程的求解方法。第1章介绍了非线性波动方程研究的历史背景,并对本文研究的方程介绍了其研究进展和研究意义。第2章简要回顾了孤立波理论的背景知识,概述了当前孤子方程求解的常见方法,最后介绍了本文研究中所需要用到的相关理论和方法。第3章在交换代数环论的基础上,利用首次积分法研究了Degasperis-Procesi方程以及带有色散项的Degasperis-Procesi方程的精确行波解,最后得到的解形式较之以前更具有一般性,扩展了该类方程解的范围。第4章考虑到参数对方程解的形式影响很大,而利用方程对应行波系统的相图可以更好地理解参数对系统解的影响,因此,用动力系统的分岔方法研究了带有色散项的Degasperis-Procesi方程的孤立波解、周期尖波解、尖峰孤立波解。分析了系统参数及奇异线对系统解结构的影响,讨论了各种行波解之间的演化过程及相互作用模式,特别的对于尖峰孤立波解的形成和机理从动力学的角度予以了讨论。第5章用动力系统的分岔方法研究了双组份Degasperis-Procesi方程的有界行波解。通过对平衡点的分析给出参数平面上对应系统的相图,由相轨道的各种类型讨论对应行波解的性质,从而给出各类解存在的参数条件,进而了解这些系统所有可能存在的有界行波解,包括孤立波解、周期波解、Breaking kink (anti-kink)波解、以及Loop解,最后借助数值模拟及数学软件给出了部分解的图像。第6章对本文所做的研究工作进行了总结,并对今后的研究方向做了展望。更多还原

【Abstract】 Nonlinear wave equation is one of the important branches in nonlinear science. As a hot topic of research in nonlinear science, study on the method for finding solutions of nonlinear wave equation has become more and more challenging. Study on exact solutions of nonlinear wave equation is helpful in clarifying the underlying algebraic structure of the soliton theory and plays an important role in reasonable explaining of the corresponding natural phenomenon and application. Because of the complicity of nonlinear wave equation, there has no systematic methodology of finding solutions now. Many effective methods which have been used so far, can not ensure the boundedness of the given solutions, we then only understand the solutions partially. In this paper, the first integral method and the bifurcation methods of planar dynamical systems are used to investigate the bounded traveling wave solutions for nonlinear wave equations, which include the Degasperis-Procesi equation, the Degasperis-Procesi equation with the dispersion term and the two-component Degasperis-Procesi equation. By making use of the method proposed, the parameter variation are analyzed for the existence of the different-type bounded traveling wave solutions. The basic content of this paper are given as following:In the first chapter, the background of nonlinear wave equation is reviewed. Then the study development and significance of our equations are introduced. Also, the research arrangement of the dissertation is given in the end of this chapter.In the second chapter, the related knowledge of solitary waves is outlined and several important methods for studying soliton solutions are introduced. Finally, the concerned concepts and theories used in this paper are introduced.In the third chapter, the exact traveling wave solutions of the Degasperis-Procesi equation and the Degasperis-Procesi equation with the dispersion term are investigated via the first integral method which is based on the ring theory of commutative algebra, and our results extend the previously known results.In the fourth chapter, it is well known that the parameters influence the types of the solution of the system greatly. Bifurcation analysis has been introduced to understand the effect of the parameters on the solutions of the system better. By using the bifurcation method of planar dynamical system, we change the DP equation with the dispersion term into the traveling wave system and draw the bifurcation of phase portraits. Then periodic wave solutions and solitary wave solutions are constructed, and their convergence is showed when parameters vary. The peaked solitary wave solutions are obtained, and the reasons for the appearance of peaked points are discussed. In the end, graph of the solution is given under some parameter conditions.In the fifth chapter, the bifurcation method of planar dynamic systems is used to investigate bounded traveling wave solutions for the two component Degasperis-Procesi equation. By using the phase portrait bifurcation of traveling wave system, solitary wave solutions, periodic wave solutions and breaking kink (anti-kink) wave solutions are discussed under different parameter conditions. And loop soliton solution is also discussed. By using numerical simulation and mathematical software, the graph of some solutions is made under specific parameter conditions.Finally, the summary of this dissertation and the prospect of study on the nonlinear evolution equations are given.更多还原