Stability Properties Of Solitary Waves For Nonlinear Schr(?)dinger Equations With Potentials

In quantum mechanics,Schrodinger equations are a class of partial differential wave equations that describe the evolution of the quantum states of a physical system over time.Since the Schrodinger equation was proposed,it has always been the frontier research content in the field of mathematical physics.In mathematics,the Schrodinger equation with the nonlinearity is called the nonlinear Schrodinger equation,which is a typical class of dispersion wave equations.Solitary wave is a special kind of solution of nonlinear Schrodinger equation,which mainly includes traveling wave and standing wave.Standing wave is generated by the equilibrium of dispersion effect and nonlinear effect of the equation.The stability properties of standing wave include stability and instability,which is one of the important criteria to measure the physical value of nonlinear Schrodinger equation,and for the nonlinear Schrodinger equation with potentials,it can reflect the physical nature more accurately.This dissertation mainly studies the existence,stability and instability of standing waves of two kinds of nonlinear Schrodinger equations with potentials,especially the ground state standing waves.The frequency of standing wave and the nonlinear index of the nonlinear Schrodinger equation are important factors to measure the properties of standing wave,so this dissertation first investigates the existence of standing wave under given conditions.Then,based on the existence of standing wave,this dissertation further investigates the stability and instability of standing wave.The specific contents are arranged as follows:1.The physical background and research status of three kinds of nonlinear Schrodinger equations related to the research contents of this dissertation are introduced,especially summarizing the research contents of the existence,stability and instability of standing waves in these equations.2.The following nonlinear Schrodinger equation with Sobolev critical nonlinearity is studied as follow:i(?)tψ=-Δψ-μ|ψ|p-1ψ-|ψ|2*-2ψ.Firstly,the variational characteristics of ground state is determined according to the corresponding elliptic equation of the equation,and a sufficient condition for the existence of ground state is obtained according to the symmetric compactness principle,thus obtaining the existence result of ground state in a certain frequency range.Secondly,a scaling transformation of the ground state is defined and an approximation relationship is established with the ground state of the classical nonlinear elliptic equation.Then combining with a sufficient condition about the stability of standing wave,the stability result of standing waves of the equation is obtained for sufficiently small frequencies in L2-subcritical case.Finally,another scaling transformation of the ground state is defined and an approximation relationship is established with the Talenti function which is the ground state of Lane-Emden equation.Then combining with a sufficient condition about the instability of standing wave,the instability result of standing waves of the equation is obtained for sufficiently large frequencies in L2-subcritical case.In addition,by using invariant sets,the strong instability result of standing waves of the equation is obtained for all positive frequencies in L2-critical and L2-supercritical cases.3.The nonlinear Schrodinger equation with harmonic potential and double power nonlinearities is studied as follow:i(?)tψ=-Δψ+γ|x|2ψ+a|ψ|p-1ψ-b|ψ|q-1ψ.Firstly,the geometry of mountain path is constructed and the variational characteristics of ground state is determined according to the corresponding elliptic equation of the equation,and by using the mountain path lemma and the weighted compact principle,the existence results of ground state are obtained in a certain frequency range.Secondly,a variational problem is defined to solve the infimum of energy functional under the constraint of mass functional,and the result that the equation has stable ground state standing waves is obtained by using the weighted compact principle in L2-sub critical and L2-critical cases of the nonlinear index q.Finally,a scaling transformation of the ground state is defined and an approximation relationship is established with the ground state of the classical nonlinear elliptic equation.Then combining with a sufficient condition about the instability of standing wave,the instability results of standing waves of the equation are obtained for sufficiently large frequencies in L2-critical and L2-supercritical cases of the nonlinear index q.