【作者】 肖奕鑫；

【导师】 唐生强；

【作者基本信息】 桂林电子科技大学 ， 应用数学， 2011， 硕士

【摘要】 在科学应用中出现的重大问题中,各种各样非线性色散方程显式精确解的研究已经引起了人们的关注.这些显式精确解的研究,采用了数学上各种分析方法,诸如反散射法,达布变换法,双线性法,李群方法,动力系统分支理论,正余弦函数方法,双曲正切函数法,Fan函数展开法,齐次平衡方法等,目前,没有统一的方法可以用来处理所有类型的非线性微分方程.在寻找非线性扩散方程精确解的各种方法中,双曲正切函数法和正余弦函数方法是求精确解最直接而有效的代数方法之一.本文将在第三章采用扩展双曲正切函数法得到(N+1)维sine-cosine-Gordon方程的显式精确扭结波解(或反扭结波解)和显式精确周期波解.本文将在第四章采用A.M.Wazwaz推广的正余弦函数方法来研究一类非线性四阶广义的Camassa-Holm方程.从而显示这种方法是一种求解非线性微分方程的有效方法.最后对本文的工作进行了总结,提出了有待于进一步解决的问题.更多还原

【Abstract】 Studies of various explicit exact solutions of nonlinear dispersive equations hadattracted much attention in connection with the important problems that arise in scientificapplications. Mathematically, these explicit exact solutions have been studied by usingvarious analytical methods, such as inverse scattering method, Darboux transformationmethod, Hirota bilinear method, Lie group method, bifurcation method of dynamic systems,sine-cosine method, tanh function method, Fan-expansion method, homogenous balancemethod and so on. Practically, there is no unified technique that can be employed to handleall types of nonlinear differential equations. The tanh method and the sine-cosine methodare ones of most direct and effective algebraic method for finding exact solutions ofnonlinear diffusion equations. In Chapter 3, the (N+1)-dimensional sine-cosine-Gordonequations are studied. The existence of kink (or anti-kink) traveling wave explicit exactsolutions and periodic traveling wave explicit exact solutions are proved, by using theextended tanh method. In Chapter 4, a class of nonlinear fourth order variant of ageneralized Camassa-Holm equation is studied by using the sine-cosine method extendedby A. M. Wazwaz. It is shown that the extended tanh method provides a powerfulmathematical tool for solving a great many nonlinear partial differential equations inmathematical physics. Finally, the summary of this thesis and the prospect of future studyare given.更多还原