某些非线性发展方程孤立波解的稳定性和动力系统的性质

【作者】 杨慧；

【导师】 郭柏灵；

【作者基本信息】 中国工程物理研究院 ， 应用数学， 2008， 博士

【摘要】 本论文首先研究刻画深水波的Benjamin-Ono方程的孤立波解和激光等离子体相互作用而得的广义Zakharov方程组孤立波解的轨道稳定性，这两个方程存在孤立波解，而且可以化为Hamilton形式(du)/(dt)=JE’(u)，其中E是能量泛函，J是一个反对称线性算子．然后利用M．Grillakis，J．Shatah和W．Strauss[21，22]提出的抽象的轨道稳定性理论以及详尽的谱分析，证明了它们的孤立波解是轨道稳定的．接着讨论描述等离子体中非线性漂移波演化过程[78，79]的Hasegawa-Mima(简记为HM)方程，主要证明了三维情形下描述静态的电子漂移波和离子声波耦合[76]的广义HM方程组周期初边值问题和Cauchy问题整体光滑解的存在性和惟一性．并给出了具粘性项ε△~2u的二维HM方程初值问题当粘性系数ε趋向零时局部光滑解趋向于相应的HM方程初值问题的解，以及收敛速度阶数的估计．本论文分为五章：第一章，介绍Benjamin-Ono方程、Zakharov方程和HM方程的研究背景和进展情况，并阐述我们的主要结果．第二章，首先阐述M．Grillakis等人提出的轨道稳定性理论，并应用该方法给出Benjamin-Ono方程孤立波解的轨道稳定性证明．第三章，应用轨道稳定性理论结合谱分析证明了广义Zakharov方程组孤立波解的轨道稳定性．第四章，用Faedo-Gal(?)rkin方法讨论三维广义HM方程组周期初边值问题和Cauchy问题解的适定性．第五章，证明了具粘性项ε△~2u的二维HM方程初值问题当粘性系数ε趋向零时局部光滑解趋向于相应的HM方程初值问题的解，以及收敛速度阶数的估计．更多还原

【Abstract】 In this dissertation we firstly consider the orbital stability of solitary waves for Benjamin-Ono equation which derived from fluids of great depth and for the generalized Zakharov equations which is the interaction of laser and plasma, respectively. There exist solitary waves of these two equations, and they can be rewritten in the following abstract Hamiltonian systems of the form(du)/(dt)=JE’(u),here E is a functional (the energy) and J is a skew-symmetric linear operator. By applying the abstract theorem of M. Grillakis , J. Shatah and W. Strauss[21, 22] and the detailed spectral analysis, we obtain their solitary waves are orbital stability .Secondly, we study Hasegawa-Mima(abrreviate HM) equation[78, 79] which describes the evolution of nonlinear drift waves in plasma, we discuss the generalized HM equations coupled electrostatic electron-drift waves and ion-acoustic waves[76] in three-dimensions, the existence and uniqueness of the global smooth solution for the periodic boundary problem and Cauchy problem are proved. And we also prove that the local smooth solution for the initial problem of HM equation with viscous termε△~2u in two dimensions can converge to the solution for the initial problem of the corresponding HM equation when the viscous coefficientεvanished, and give the estimate for the order of the convergent speed.The dissertation consists of five chapters:In chapter 1, we briefly introduce the background in physics and developments of Benjamin-Ono equation, Zakharov equation and HM equation, in addition the mam results of the dissertation is described.In chapter 2, we interpret the abstract orbital stability of M. Grillakis et. al, and prove that solitary waves of Benjamin-Ono equation is orbitally stable.In chapter 3, by applying the abstract stability theorem and detailed spectral analysis we obtain the orbital stability of the solitary waves for the generalized Zakharov equations. In chapter 4, we consider the well-posedness of the periodic boundary problem and Cauchy problem for the generalized HM equations in three dimensions.In chapter 5, we study HM equation in two dimensions, and prove that the local smooth solution for the intial problem of the HM equation with viscous termε△~2u can converge to the solution for the initial problem of the corresponding HM equation when the viscous coefficientεvanished, and give the estimate for the order of the convergent speed.更多还原