A Study On The Solution To The System Of Two-dimensional Cubic Nonlinear Schrodinger Equations With Potentials

Schr(?)dinger equation is considered as one of the foundational theories of quantum mechanics,it reveals the basic law of matter motion in the microphysical world.As a typical dispersion equation with strong applications in nonlinear optics,the research on the related properties of the solution to nonlinear Schr(?)dinger equation has attracted much attention.In the study of the related properties of the solutions of the nonlinear Schr(?)dinger equation,the time decay estimates and asymptotic behavior of the solution are basic problems,which are generally discussed by the energy methods and Strichartz estimates.This paper will elaborate the work done through four parts.The first part systematically explains the reason and purpose of the work by introducing the research background and significance of nonlinear Schr(?)dinger equation and the research status of nonlinear Schr(?)dinger equations at home and abroad;The second part mainly introduces some basic symbols,fundamental concepts and inequalities,so as to provide a theoretical basis for the later proofs;The third part,which is also an important part of this paper,studies the initial value problem of the following two-dimensional cubic nonlinear Schr(?)dinger equations with potential:where F1(v1,v2)=|v2|2v2,F2(v1,v2)=|v2|2v1,vj(t,x)is a complex valued unknown function,Δwj=Δ-Wj(x),Wj(x)is a real valued function on R2,vj is the complex conjugate of vj,mj is the particle mass,j=1,2.In order to consider the time decay estimate of the solution of this special two-dimensional cubic nonlinear Schr(?)dinger equation,we give some appropriate assumptions on the potential function Wj(x).Firstly,the local existence of the solution is proved by the principle of contractive mapping principle,and then the existence of the global solution is proved according to a prior estimates of the local solution;Secondly,through the proof of relevant lemma,the time decay estimate of the solution is further analyzed by using commutative properties,factorization and harmonic analysis methods;Finally,if the initial value‖φj(x)‖Hα∩H0,α(1<α<4/3)is sufficiently small,the nonlinear Schr(?)dinger equations studied are asymptotically complete;In the fourth part,the special two-dimensional cubic nonlinear Schr(?)dinger equations in the third part are extended to a more general form,and then the initial value problem of the following two-dimensional cubic nonlinear Schr(?)dinger equations with potential is studied:where Fj(v1,…,vh)=∑1≤i≤k≤l≤2h λi,k,ljvivkvl,λi,k,lj∈C,and there is a positive number cj,so that Im ∑j=1h cjFjvj=0,ΔWj=Δ-Wj(x),Wj(x)is a real valued function on R2,vj(t,x)is a complex valued unknown function,vj is the complex conjugate of vj,mj is the particle mass,j=1,2,…,h.By using the properties of mass resonance,the Strichartz estimates,the properties of Schr(?)dinger operator and a priori estimates of local solutions,the global existence of the solution of the above equations is proved,the long-time asymptotic behavior of the solution is analyzed,and the scattering problem of the equations is discussed.