【作者】 刘恂；

【导师】 田立新；

【作者基本信息】 江苏大学 ， 系统工程， 2010， 博士

【摘要】 非线性现象是自然界最普遍的现象,是自然界的本质.非线性系统的提出和研究,促使不同学科相互渗透融会,大批新兴学科应运而生,逐步诞生了探讨复杂性现象的非线性科学。非线性科学主要包括研究孤子理论、混沌理论、分形理论和耗散结构理论等等,以及这些理论在其他相关学科领域的广泛应用。孤波理论与应用是非线性科学研究的热门课题之一。孤立子相关性质的研究在揭示波的传播规律、准确解释自然现象和科学应用相关技术等方面均具有极大的科学研究价值.非线性发展方程的研究又离不开孤立子理论.大量的非线性问题的研究和解决最终都归结为求解非线性偏微分方程(组)的问题.非线性偏微分方程(组)的求解要远比线性偏微分方程(组)的求解困难得多,很难用统一的方法对前者加以处理.由此,求非线性偏微分方程(组)精确解的工作,就显示了很重要的理论和应用价值.本文基于此目的,在归纳和总结了现有各种主要的精确求解非线性发展方程方法的基础上,研究了一类具有实际应用物理背景的非线性波动方程,如mBBM方程、MCH方程,Klein-Gardner方程,组合KdV-mKdV方程、广义的BBM方程等.把一些经典的研究方法加以推广和改进,借助于符号计算和数学机械化的方法,来研究这类非线性波动方程的行波解,不但获得了已有的结果,而且得到了一些新的结果.通过研究波动方程的动力学性质,从定性角度数形结合分析,寻求非线性波动方程的解.全文共分七章.第一章介绍了非线性波动方程提出的研究背景、进展和现状,提出了本课题的研究意义和研究内容.第二章介绍了几个重要的求解非线性波动方程精确解的方法,并简要阐述了本文主要的求解非线性波动方程的方法以及与本文相关的基本概念和基本原理.第三章借助于首次积分法,对常见的mBBM方程、简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,进行了全面的分析,得到了一些精确解,并结合简单的直接积分,以及Jacobi椭圆正弦函数展开法,比较了这个方法的优点。第四章应用(?)展开法,对以上讨论的简化形式的MCH方程,Klein-Gardner方程,组合KdV-mKdV方程,再次作进一步的研究,得到了这些方程的双曲函数形式、三角函数形式的解,丰富了这些方程的讨论。这类方法的应用日益丰富,甚至对一些重要的离散的孤波方程也一样适用.第五章借助Wazwaz的独创性的工作,结合著名的Hirota法,用一种简单的方式,有效地寻求了(2+1)维Zakharov-Kuznetsov方程(简记为ZK(m,n,k))、破裂孤子方程、Potential Kadomtsev-Petviashvili (PKP)方程以及一个五阶色散方程的解,并且包含多重孤子解和奇异孤子解。本文还尝试利用同宿测试法(homoclinic test method),讨论Hirota法的一些延伸工作。第六章受启发于Kuru等人利用二阶微分算子的分解理论,直接将Wazwaz等人讨论过的几类广义BBM方程以及两个修正的Boussinesq方程,经过行波变换后,通过Weierstrass椭圆函数的形式,求得这几类方程的行波解,主要是周期解和双曲函数解,且多数解的形式未曾在文献中被发现。最后,结合已有的一些结论,对各类方法的后续展开,作了初步的展望,并为以后的尝试提供平台。更多还原

【Abstract】 Nonlinear phenomenon is the most common phenomenon in nature, and it is the essence of nature. When nonlinear systems were proposed and researched, it promoted fusion of different disciplines. A large number of new disciplines emerged, and gradually gave birth to the nonlinear science to systematically study the complexity of the phenomenon. Nonlinear science includes the research of soliton theory, chaos theory and fractal theory and theory of dissipative structures, etc., and the extensive application of these theories in other related disciplines. Solitary wave theory and the application is a hot topic in nonlinear science.The related properties of soliton play great importance in revealing the wave propagation, in the accurate and scientific explanation of natural phenomenon, and in the related application of engineering techniques. In order to describe these, the nonlinear evolutionary equation (or equations) was introduced, and therefore there are both theoretical and practical meanings both in quantitative research and qualitative research. The research of amount of nonlinear problem comes to the research of the nonlinear evolutionary equation (or equations).However, it is more sophisticated to solve the nonlinear equation than to solve linear one, and generally there isn’t a unified approach to deal with the former. Thus, it shows a very important theoretical and practical value for solving nonlinear partial differential equation (or equations).Based on this, in the summary of the major existing methods for solving nonlinear evolution equations, the paper applies several methods to some typical equations, such as mBBM equation, MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, generalized BBM equations. From the application, the paper got some old solutions and some new ones.The full-text is divided into seven chapters. The first chapter describes the research background, progress and nowadays works of nonlinear wave equation.The second chapter introduces several important methods for looking for exact solutions of nonlinear wave equation, and briefly describes the basic concepts and principles in this paper for solving nonlinear wave equation associated with this article.In the third chapter, by means of the first integral method, I analyzed some equations, such as mBBM equation, simplified form of MCH equation, Klein-Gardner equation, and combined KdV-mKdV equation. By a comprehensive analysis, I find some exact solutions, combined with a simple direct integration and the expansion method of Jacobi elliptic sine function.In the next chapter, I apply (G’/G) expansion method to discuss the simplified form of MCH equation, Klein-Gardner equation, combined KdV-mKdV equation, and get many solutions with the forms of hyperbolic function, trigonometric function, which enrich the discussion of these equations. The increasingly rich application of the method is still suitable for some important discrete solitary wave equation.In the fifth chapter, inspired by Wazwaz’s original work with well-known Hirota method, I apply a simple way to effectively search for the solutions of the (2 +1)-dimensional Zakharov-Kuznetsov equation (abbreviated as ZK (m, n, k)), breaking soliton equation, Potential Kadomtsev-Petviashvili (PKP) equation, and a fifth-order dispersion equation, and includes multiple soliton solutions and singular soliton solutions. Still I attempt to utilize the homoclinic test method to discuss the extension work of the Hirota method.In the sixth chapter, I am enlightened by the work of Kuru’s decomposition theory of second order differential operator, and discuss several classes of generalized BBM equations which were researched by Wazwaz and two modified Boussinesq equations. After traveling wave transformation, with the Weierstrass function form, the paper obtains some types of traveling wave solutions, among which are periodic solutions and hyperbolic function solutions, and most of the solution has not been found in the literature.Finally, there are some of the conclusions of the various methods. I make some summary and forecast. Based on these, I would make future attempts.更多还原