Nature Of Some Of The Stability Of Solitary Wave Solutions Of Nonlinear Evolution Equations And Dynamical Systems

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.