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-展开法及其非线性演化方程的孤立波解
目录
摘要 II
Abstract II
第1章绪论 2
非线性偏微分方程 2
非线性科学概述 2
非线性偏微分方程的解法综述 2
数学机械化与符号计算 2
2
孤立子及尖峰孤立子 2
孤立子分类 2
第2章方法介绍 2
齐次平衡原则 2
-展开法 2
-展开法简介 2
扩展的-展开法 2
3 研究实例 2
-展开法的方程求解 2
2
常系数的KDV方程 2
-展开法的方程求解 2
-展开法的方程求解 2
-展开法的方程求解 2
结语 2
参考文献 2
-展开法及其非线性演化方程的孤立波解
摘要
,如何求解非线性数学物理方程在现代科学研究中具有重要的理论和实践价值.
本文在第一章中简要介绍了非线性偏微分方程的发展及孤立子的基本概念。在第二章中介绍了王明亮、李志斌教授的齐次平衡原则和展开法,重点是对展开法的拟解形式做出了三种改进和拓展。经典拟解和拓展拟解形式为是: 、、和。在第三章中运用第一种经典拟解形式求解了修正的常系数Kawachara方程和常系数KDV方程,得到了多种形式的孤立波解;运用第一种改进的拟解形式求解了常系数BBM方程,也得到了各种形式的孤立波解;运用第二种改进的拟解形式求解了常系数色散耗散方程,得到了各种形式的孤立波解;运用第三种改进的拟解形式求解了变系数Burgers方程,也得到了相应的三角函数和指数形式的孤立波解。
关键词:-展开法;齐次平衡原则;非线性偏微分方程;孤立波解
-expansion method and solitary wave solutions of onlinear evolution equations
Abstract
The progress in physics depends heavily on nonlinear mathematics and the progress of solving nonlinear equation method. Therefore, how to solve the nonlinear equations of mathematical physics has important value of theory and practice in modern scientific research.
In the first chapter ,This article briefly introduced the development of nonlinear partial differential equations and the basic concepts of soliton. In the second chapter, this article briefly introduced homogeneous balance principle proposed by Wang Mingliang and Professor Li Zhibin and the expansion method. It is important to make the three forms of the Improved quasi-solution of the expansion method. the forms of the classic quasi-solution and Improved quasi-solution are:
、and. In the third chapter ,this article solves a amendment constant coefficients Kawachara equation and the constant coefficients KDV equation by the classical form of the expansion quasi-solution,and obtains Various forms of solitary wave solutions. this article solves the BBM equation with constant coefficients by the first improved for