Orbital Stability Theory Of Solitary Wave Solutions For Two Types Of Derivative Nonlinear Schr?dinger Equation

Schr?dinger equation is an important equation of partial differential equation and plays an fundamental role in the quantum mechanics equation.In this thesis,we study the orbital stability theory of solitary wave solutions for two types of derivative nonlinear Schr?dinger equations.The nonlinear Schr?dinger equation of derivative type is written as(DNLS-b):there exists a two-parameter family of solitary wave solutions of the form:where b?0,(?,c)??:={(?,c)?R+×R:c2<4? or c = 2(?)}.Moreover,??,c satisfies:Ohta[92](2014),proved that for b>0,there exists ? = ?(b)?(0,1)such that:the solitary wave solutions u?,c(t,x)of(0.0.3)is stable when-2(?)<c<2?(?)and unstable when 2?(?)<c<2(?).After this work,the stability theory in the endpoint case c = 2(?)and the degenerate case c =2?(?)remain open.In Chapter 2,we show that for b>0,the solitary wave solutions u?,c(t,x)of(0.0.3)is unstable in the endpoint case c =2?(?).In the endpoint case,the two-parameter family of solitary wave solutions becomes one-parameter family.We still want to construct a"negative direction" which satisfies two orthogonal condition according to one parameter,this is the diffculty in this chapter.In order to construct the "negative direction",we use localization technique and the special properties of the equation and the solution.Besides,combining with the Lyapunov arguments,finally we showed the instability.In Chapter 3,we proved that for b>0,the solitary wave solutions u?,c(t,x)of(0.0.3)is unstable in the degenerate case c =2?(?),where ?=?(b)?(0,1),such that P(??,c)=E(??,c)=0.In the degenerate case,the Lyapunov arguments don't work any more.Our proof was based on the modulation argument which was introduced by Weinstein[110]and extended by Martel and Merle[75],and the virial identities which was obtained by Wu[113].Finally,we proved the instability.The generalized derivative nonlinear Schr?dinger equation is as following(gDNLS):there exists a two-parameter family of solitary wave solutions of the form:where ?>0,(?,c)? ? and ??,c is the solution of:Liu,Simpson and Sulem[73](2013),proved that for 1<?<2,there exists z0(?)?(0,1)such that:the solitary wave solutions u?,cg(t,x)of(0.0.4)is stable when-2(?)<c<2z0(?);and unstable when 2z0(?)<c<2(?).Further,Fukaya[29](2016),proved that the solitary wave solutions u?,cg(t,x)is unstable when 7/6<?<2,c = 2?(?).After this work,the stability theory when c =2?(?)(1<??7/6)and c =2(?)are unsolved.In Chapter 4,we proved when 1<?<2,the solitary wave solutions u?,cg(t,x)of(0.0.4)is unstable in the degenerate case c =2z0(?).In the degenerate case,Lyapunov arguments need high-order regularities.In order to avoid the requirement,using the similar ideas in Wu[115],we constructed a kind of virial functionals instead of Lyapunov functionals.Combining with the modulation,we showed the instability.