【作者】 李春海；

【导师】 黄文韬；

【作者基本信息】 桂林电子科技大学 ， 应用数学， 2010， 硕士

【摘要】 非线性波方程是描述自然现象的一类重要数学模型,也是非线性数学物理特别是孤立子理论最前沿的研究课题之一.通过对非线性波方程的求解和定性分析的研究,有助于人们弄清系统在非线性情况下的运动变化规律,合理解释相关的自然现象,更加深刻地描述系统的本质特征,推动相关学科如物理学、力学、应用数学以及工程技术的发展.随着非线性科学的发展,许多物理、化学和生命科学模型都可以转化为非线性方程,如非线性常微分方程、偏微分方程等.非线性方程的求解已经成为非线性科学领域的一个重要研究课题.本文主要利用积分因子方法和动力系统分支理论研究了几类非线性波方程的行波解及性质,并进一步研究了一类奇异扰动非线性波方程孤立波解的存在性.全文共有六章组成.第一章是绪论,对非线性波方程的发展历史、研究现状、研究意义进行了叙述.第二章是预备知识,主要介绍了与本文相关的一些基础理论和方法.在第三章,我们用积分因子方法研究了两类非线性波方程,广义Camassa-Holm方程和广义G-P程,求出了它们的孤立尖波,孤子类解和周期解.在第四章,我们利用动力系统分支理论研究了一类广义双sinh-Gordon方程和一类(N+1)维sine-cosine-Gordon方程,讨论了它们的相图及其分支,给出了明确的参数条件以及参数条件下的相图.并给出了行波解的精确参数表示.在第五章,我们应用几何奇异扰动定理研究了一类奇异非线性波方程,对奇异扰动mKdV方程孤立波解的存在性进行了证明.最后,就全文进行了总结,就研究中还没有彻底解决的问题进行了说明,并提出了有待进一步研究的问题.更多还原

【Abstract】 Nonlinear wave equations are important mathematical models for describing nat-ural phenomena and are one of the forefront topics in the studies of nonlinear math-ematical physics, especially in the studies of soliton theory. The research on findingand analyzing exact solutions of nonlinear wave equations can help us understand themotion laws of the nonlinear systems under the nonlinear interactions, explain thecorresponding natural phenomena reasonably, describe the essential properties of thenonlinear systems more deeply, and greatly promote the development of engineeringtechnology and related subjects such as physics, mechanics and applied mathematics.With the boom of nonlinear science, many models in physics, chemistry and lifesciences can be converted into nonlinear equations, such as nonlinear ordinary dif-ferential equation and partial di?erential equation. Consequently, solving nonlinearequations has become an important research topic in the field of nonlinear science.In this thesis, we investigate several types of nonlinear wave equations by usingthe technique of integral method and bifurcation theory of planar dynamical systems,calculate their travelling wave solutions, and further study on the existence of solitarywave solutions of perturbed nonlinear wave equation. This paper is formed by sixchapters.In Chapter 1, the historical background, research developments and significanceof nonlinear wave equations are summarized.In Chapter 2, the basic theory and method of nonlinear wave equation are pre-sented.In Chapter 3, by using the technique of integral factors, the peakons, solitarypatterns and periodic solutions of generalized Camassa-Holm equation and generalizedG-P equation are obtained.In Chapter 4, we investigate the generalized double sinh-Gordon equation and(N+1)-dimensional sine-cosine-Gordon equation by using the bifurcation theory of pla-nar dynamical systems, discuss and analyze their phase portrait and branches, then,work out the exact solutions of the equation.In Chapter 5, by taking advantage of singular perturbation theory, the existenceof solitary wave solutions of perturbed mKdV equation is proved.Lastly, a summarization of the whole paper and the still unsolved problems in theresearch are given. Moreover, the future study is prospected.更多还原