Orbital Stability And Instability For Several Types Of Nonlinear Waves

In this dissertation, combining GSS method, classical method with detailed spectral analysis, we mainly study the orbital stability and instability of the soliton solutions and periodic traveling wave solutions of the elliptic function for some nonlinear dispersive partial differential equations. These equations appear in the modern mathematical physics.The first chapter mainly introduces the research methods of orbital stability,research status for some nonlinear partial differential equations, and gives the main research content and purpose of this dissertation. In second chapter, by applying the orbital stability theory presented by Grillakis et al. and detailed spectral analysis, we prove the orbital stability and instability of six kinds of solitary wave solutions with zero asymptotic value and nonzero asymptotic value for the coupled compound KdV and MKdV equations. Chapter 3 firstly shows the existence of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the coupled nonlinear wave equation. Furthermore, we use Lam′e equation and Floquet theory to give spectral properties for some linear operators, and combine the orbital stability theory presented by Grillakis et al. to show the orbital stability of the dnoidal type periodic waves solutions with period L. In chapter 4, by using the orbital stability theory presented by Grillakis et al. and spectral analysis method developed by Lopes, we show the orbital stability of sech2-type solitary wave solutions for Klein-Gordon-Schr¨odinger equations. Chapter 5 and Chapter6 respectively prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Long-Short wave equations and the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that the periodic wave solutions for the generalized Long-Short wave equation and the generalized Zakharov equations are orbitally stable by perturbations with period L.The work of this dissertation not only improve and complement the existing orbital stability results, but also apply these research methods to study the orbital stability and instability of solitary wave solutions with nonzero asymptotic value,as well as the the orbital stability of periodic traveling wave solutions for the generalized nonlinear dispersive partial differential equations.