对某类变系数非线性发展方程精确解的讨论

【作者】 王燕；

【导师】 孙福伟；

【作者基本信息】 北方工业大学 ， 应用数学， 2010， 硕士

【摘要】 本文以探究非线性发展方程精确解的孤子理论为依据,在导师孙福伟教授的指导下,研究了在诸多自然科学领域有着广泛应用的低维变系数非线性发展方程与高维变系数非线性耦合方程求解精确解尤其是精确孤立波解的方法。论文第一章介绍国内外非线性科学中孤立子理论研究的历史与发展,重点简述求解非线性发展方程精确解的几种主要方法及该方法在国内外的研究现状。论文第二章第一部分详述Painleve分析方法在国内外的发展情况和用它求解非线性发展方程精确解的具体步骤,第二部分将Painleve分析方法用于研究非线性发展方程精确解的前沿领域之一：变系数非线性发展方程,以变系数KdV-Burgers方程为实例,求解变系数KdV-Burgers方程的精确解表达式及Backlund变换,并通过形象的、简单易行的图示描述由此变系数非线性发展方程所确定的精确孤立波解。论文第三章在理解第二章研究成果的基础上,继续探究非线性发展方程应用的高新技术领域：变系数、高维、耦合系统,继续用Painleve分析方法求解高维变系数非线性耦合方程的精确解,以有背景相互作用的2+1维变系数非线性Schodinger耦合系统为典型,求解出该变系数高维耦合系统的精确解表达式及Backlund变换,分析由此高维变系数耦合方程所确定的精确解,同样以图示方式阐释其中的精确孤立波解,深化用Painleve分析方法求解高维、变系数、耦合非线性系统精确解的研究。论文第四章总结研究成果,分别指出第二章、第三章用Painleve分析方法处理变系数、高维、耦合等这些非线性发展方程研究中的复杂问题时的物理理论根源,解释据此所采取的数学解决方法的变通与改进,分析在综合以上复杂问题时所能确定出的精确孤立波解的物理意义,延伸求解非线性发展方程精确解中对孤立子理论的探讨,挖掘课题研究中的独立创新及缺点不足,展望非线性科学发展的前景。更多还原

【Abstract】 Under the guidance of Professor Sun Fuwei, this paper explore the exact solutions of nonlinear evolution equations based on the theory of soliton, It mainly studied a wide range of low-dimensional variable coefficient nonlinear equations and high-dimensional variable Exact solutions of linear coupled equations especially exact solitary wave solutions in many natural sciences applications.The first chapter introduces the domestic and foreign history and development of non-linear science, soliton theory focusing on brief exact solutions of nonlinear evolution equations in several major ways and relate methods of research status at home and abroad.The first part of the second chapter details the Painleve analysis of development at home and abroad and specific steps of using it gain exact solutions of nonlinear evolution equations. By the Painleve analysis method, the second part study variable coefficient nonlinear equations which is one frontier of the research of nonlinear evolution equations. For variable coefficient KdV-Burgers equation as an example, the paper get the variable coefficient KdV-Burgers equation and Backlund transformation expression and describe the nonlinear evolution equations by the exact solitary wave solutions determined by variable coefficients through simple image.In the basis of understanding research results in chapter II, the third chapter continues to explore the application of nonlinear evolution equations in high-tech areas:variable coefficient, high dimension, coupled system. With the Painleve analysis method, the paper has again studied high-dimensional nonlinear coupled coefficient equation. For 2+1 dimensional variable coefficient nonlinear Schodinger system as typical of the variable coefficient problems, it gains the exact solutions of expression and the Backlund transformation of high-dimensional coupled system and makes analysis of the exact solution that determined by this high-dimensional variable coupled equations, In the same way of the second chapter, it explains exact solitary wave solutions with images, which deepen with the Painleve analysis method to solve exact solutions of high-dimensional, variable coefficients, nonlinear coupled systems research..The forth chapter summarizes the research conclusions pointed out the complex issues of physical theory causes that explain accordingly adopted Painleve analysis approach with variable coefficient, high dimension, nonlinear evolution coupled equations in chapterⅡ, ChapterⅢ.The paper identifies the modifications and improvements in the above analysis and the physical meaning of exact solitary wave solutions of complex problems, extending the exact solutions of nonlinear evolution equations on the theory of solitons, At last, it mines independent innovation and weaknesses during the research and looks forward to the prospects for the development of nonlinear science.更多还原