【作者】 申建伟；

【导师】 徐伟；

【作者基本信息】 西北工业大学 ， 应用数学， 2006， 博士

【摘要】 本文从动力系统分岔理论的角度来研究非线性波方程的行波解，行波解的分岔及其动力学行为，并结合计算机符号代数的方法和相图分析的方法给出了不同波方程可能存在的行波解的种类，分析了这些复杂行波解产生的原因。在实际模型中，有界行波解具有很强的实际应用价值，目前求解行波解的方法给出的解不能明确给出该行波解是否有界，本文根据动力系统理论的特点，利用连接平衡点的闭轨线的特点结合轨线与行波之间的对应关系来研究非线性波方程的精确行波解的显式表达式。对于具有耗散项的非线性波方程，有些情况下，精确行波解不易求出，本文就根据平衡点附近轨道的性质，给出其近似解的表达式，拓广了求解的方法。在研究非线性波方程中，很多方程会出现非解析解(非光滑解)，对于这些非解析的行波解，尤其是Peakon解和Compacton解，在文中解释了这些非解析波存在的原因，并证明了Peakon解是广义导数意义下的广义解而非弱解，Compacton解是广义导数意义下的弱解，利用分岔理论揭示了这些非解析波解与解析波解之间的关系以及产生这些行波解的分岔条件。另外本文还证明了在积分常数不为0的情况下，非线性波方程也可以产生Compacton解，当方程的首次积分比较复杂时，尤其是出现超越函数(例如对数表达式)表达式时，方程的解非常复杂，可能出现不可数多的Compacton解，这些Compacton解有一个能量界。最后，本文研究了柱面波方程的行波解及相关的分岔行为和动力学性质，揭示了在柱面系统中，存在旋转的周期波族和破缺波，给出了这些波存在的分岔条件，并解释了这些波产生的原因。更多还原

【Abstract】 The Nonlinear wave equation and bifurcation theory of dynamical system arethe hotspot in the nonlinear science, more and more scientists throw theirself intothese areas. In the paper, from the viewpoint of dynamical system, we investigatedthe traveling wave solutions, the bifurcations of traveling wave solutions and theirdynamical behavior of the nonlinear wave equation. In the mean time, combining thesymbolic algebra of computer and analysis of phase portraits, we obtained thedifferent types of traveling wave solutions between different equations and analyzedthe reason why these complex traveling wave solutions arise. In the real model,traveling wave solutions with boundary have strongly practical value, but the methodwhich find the traveling wave solutions can not ensure whether the solutions isbounded. In this paper, according to the theory of dynamical system, we investigatethe explicit and exact traveling wave solutions of nonlinear wave equation by usingcharacter of the closed trajectory connecting equilibrium points and the relationbetween the obits and traveling wave.When the wave equation has dissipative term, in some case, the exact travelingwave solutions can not be obtained, so we began to study the approximate methodwhich find the solutions. In the chapter 3, by using the dynamical behavior of orbitsnear the equilibrium points and the method of symbolic algebra of computer,omitting some higher order term, we can obtain the explicit expression ofapproximate solutions, so these enrich the method which study the approximatesolutions.When we study the nonlinear wave equations, a lot of wave equation existnon-analytical (non-smooth) solutions. For these non-analytical traveling wavesolutions, for example, especially, Peakon and Compacton, we explain the reasonwhy these non-analytical traveling wave solutions arise, and prove that peakon isgeneralized solution in the sense of generalized differential and the compacton is weak solution. In the meantime, we open out the relation between the non-analyticalwave solutions and analytical wave solutions, and the condition which bifurcation ofdifferent traveling wave solution arise.Moreover, in this paper, we prove that the nonlinear wave equation can givebirth to compacton when the integral constant is not 0. Especially, when the firstintegral exists logarithm function, the solutions of wave equation are very complex,and the wave equation may produce non-numerable compacton which the energy ofthese compacton have a maximum.Finally, we investigate the traveling wave solutions, bifurcations of its andthe their dynamical behavior of the hydrogen-bounded system in the phase cylinder,show that there exists rotation periodic wave solutions and breaking wave anddisplay the bifurcation which these traveling wave arise. In the paper, we alsoexplain these waves arising.更多还原