Asymptotic Behavior Of Solutions For A System Of Nonlinear Schrodinger Equations Under Long And Short Range Interactions

The nonlinear partial differential equation is an important branch of modern mathematics.In recent years,many significant problems in natural science,mechanics,engineering science can be attributed to the nonlinear partial differential problem.The nonlinear Schrodinger equation is a staple part of nonlinear partial differential equations,and the relevant research is one of the indispensable nonlinear problems that have been studied by many mathematicians and physicists.Many scholars have studied the nonlinear Schrodinger equation theory,and also made a lot of achievements.This paper consists of four chapters.Firstly,it introduces the historical back-ground,relevance of significance and research status for the nonlinear Schrodinger equation,and summarizes the main contents of this paper.Secondly,we introduce the basic concepts needed in this paper,and then give some classic results.Thirdly,this paper mainly studies the small initial value problem for our target system.We prove the global existence of solutions and show the time decay of solutions to the under long-short range interactions.We decompose the nonlinear terms of the Schrodinger system by the mass resonance condition and factorization method.The time decay es-timates are obtained by using an appropriate operator,the energy method and Sobolev inequalities.Then we introduce the initial value problem of the generalized nonlinear Schrodinger system,Finally,this paper mainly studies the existence of global solutions and the long time asymptotic behavior of the one-dimensional nonlinear Schrodinger system under long-long range interactions.It can be divided into two cases:1.The small initial value problem of a strongly dissipative system.By the mass resonance condition and factorization method,we have an ordinary differential system.The time decay estimates of the system are obtained by using the solution of the corresponding free system,energy method,strong dissipative condition,Sobolev inequalities and Young inequalities.2.The small initial value problem of the dissipative system.we restrict the second nonlinear coefficients of the system.Using the dissipative condition,we get the time decay estimates based on the proof methods of the strongly dissipative case and the nonlinear system under long-short range interactions.