The Behaviour Of Solutions In Some Nonlinear Schr(?)dinger Equations

The motion law of microscopic particles has been widely studied via nonlinear Schr(?)dinger equations and their solutions.Study on Schr(?)dinger equations and the solutions has been one of the most significant breakthroughs in quantum mechanics since the 20 th century.As one global popular topic,it has received much attention from physics?mathematics?biology?geography and so on.Based on the above reasons,this dissertation focused on studying dynamic behaviour of solutions in some nonlinear Schr(?)dinger equations.In Chapter 1,we give a brief introduction on the historical background,recent advances of research on nonlinear Schr(?)dinger equations.Our main results are also outline in this chapter.In Chapter 2,we study the existence and orbital instability of standing waves for Klein-Gordon-Schr(?)dinger system with quadratic-cubic nonlinearity.By variational methods we first show the existence of ground states.Then we establish a Virialiden-tity for this system,which,together with the Virial theorem,allows us to prove that the standing waves we obtained are orbitally instable if the frequency?is sufficiently small.Our results improve and complement some previous ones.In Chapter 3,using the(G?/G)-expansion method,we obtain some exact solutions for a coupled discrete nonlinear Schr(?)dinger system with a saturation nonlinearity.These exact solutions include the hyperbolic function type,trigonometric function type and rational function type,of which the hyperbolic function type solutions may generate kink and anti-kink solitons.In Chapter 4,the discrete solitons are considered from the anti-continuum limit of the nonlinear Schr(?)dinger equation with saturable nonlinearity.We first calculate the exact interval of the dispersion coefficient?.Some sufficient conditions are given for the existence of discrete solitons,which are exponentially decaying.Secondly,taking the discrete exact solution into consideration,a representative numerical sketch of some different kinds of solitons are presented in the paper.By the regular perturbation theory,spectrum stability and instability results for discrete solitons in the anticontinuum limit are established.Finally,we show the asymptotic approximations for the eigenvalues of the linearized stability problem.A summary of this dissertation and the outlook for future research work are stated in Chapter 5.