摘要
非线性现象是自然界中既普遍又重要的现象。非线性科学是研究非线性现象共性的一门学问,它研究的主体是孤立子,混沌和分形。许多非线性问题的研究最终可归结为非线性系统的描述。
非线性系统的精确解对研究相关的非线性问题非常重要。孤立子研究的一个主要内容,就是寻求非线性系统的解,特别是孤立波解。在过去大约50年中,非线性科学研究领域颇具特色的新成果之一就是创造了求非线性方程的解特别是孤波解的各种精巧方法。
本文正是应用上述方法来研究非线性的色散波方程的孤立子解。在第一、二章首先介绍了非线性波动方程及孤立子理论的研究背景、研究进展和发展现状和意义,总结并分析了现有的求解非线性波动方程的方法。随后介绍了本文研究非线性波动方程孤立波解所用的方法及涉及的相关的概念、定理。
第三章,利用动力系统的定性分析理论,通过相图分析的方法,借助Mathematical软件,通过同宿轨,异宿轨,周期轨,对应解的情况,研究了广义CH方程的孤立子解,并且给出了光滑周期波解,光滑孤立波解,类似扭波解和类似反扭波解的存在条件以及周期尖波解,孤立尖波解的精确表达式。第四章,利用相图分析的方法,研究了Fornberg-Whitham方程的类似扭波解和类似反扭波解,给定参数值,求出了解的精确表达式,画出其图形。
Nonlinear Science,which has solution theory,fractal and chaos as its main parts,is the subject of studying the common futures of nonlinearity. Nonlinearity is universal and important.Most nonlinear problem can be described by nonlinear equations,which generally includes nonlinear.
How to obtain the exact solutions of nonlinear equations is of vital importance to study of the corresponding problem.The key problem in soliton theory is to get solutions of the nonlinear evolution equations, including exact ones or numerical ones.During the past 50 years or so, the scientists have created various ingenious methods to construct exact solutions,especially soliton solutions of nonlinear equations.
In this paper,we should used above methods to study nonlinear dispersive wave equation.In chapter 1 and chapter 2,we introduce the study background,study development and significance of nonlinear wave equation and soliton theory.The methods known up to today for solving the nonlinear wave equation are summarized and analyzed.Then the concerned concepts and theories which used in this paper are introduced.
In the third chapter,the qualitative analysis methods of dynamical system are used to investigate the soliton solutions of the generalizedγ-CH equation.Using the way of the phase portrait bifurcation analysis and mathematical software,we obtain the soliton solutions by the solutions corresponding to the homoclinic orbit, heteroclinic orbit and the periodic orbit.However we give the conditions of there are exist smooth periodic wave solutions,smooth soliton wave solutions,periodic cusp solutions,kink-like wave solutions,antikink-like solutions.And exact explicit parametric representations of periodic cusp solutions and soliton cusp wave solutions are given and the numerical simulation is made.In the forth chapter,using the way of the phase portrait bifurcation analysis,we study kink-like wave solutions and antikink-like solutions of the F-W equation.The exact explicit parametric representations of those solutions are obtained and the numerical simulation is made.
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