Study On Soliton Decomposition Of Nonlinear Schr(?)dinger Equations

Modern Mathematical physics has two major pillars:Einstein’s theory of relativity and quantum mechanics.Schr(?)dinger equation is the basic model of quantum mechanics,which describes how the quantum state evolves with time.Nonlinear Schr(?)dinger equations are concerned with the nonlinear interactions in the quantum states.The research on the evolution laws of nonlinear Schr(?)dinger equations has always been a hot topic in mathematics and physics.The soliton decomposition conjecture of the nonlinear Schr(?)dinger equation refers to the global solution of the equation that can asymptotically decompose into the sum of multiple solitons and an infinitesimal remainder in energy space.This problem is a significant open problem in the field of modern Mathematical physics.After Terence Tao,the winner of the Fields Medal,put forward it clearly in 2009,it has caused the vigorous rise of multi soliton structure and related research.The multi solitons refer to the global solution of an equation that can be infinitely approximated by the sum of multiple solitons.Obviously,the research on multi soliton decomposition is in line with the soliton decomposition conjecture.Small solitons refer to solitons with a mass less than a precised value.Compared to general solitons,small solitons have many special properties,especially in terms of stability.This dissertation mainly studies the orbital stability of small solitons in two types of nonlinear Schr(?)dinger equations,the essential correspondence between prescribed mass and soliton frequency in the normalized solution problem,multi solitons constructed by stable solitons,and the existence of high speed excited multi solitons for a class of nonlinear Schr(?)dinger equations.The main contents of this dissertation are as follows:1.This dissertation studies the existence and orbital stability of small solitons for nonlinear Schr(?)dinger equation with double power nonlinearity,the essential correspondence between the prescribed mass and soliton frequency in the normalized solution problem corresponding to small solitons,and the multi soliton properties of equations constructed by stable small solitons.Firstly,by using the Gagliardo-Nirenberg inequality,this dissertation gets the global well posedness of the nonlinear Schr(?)dinger equation.Secondly,in terms of the Cazenave-Lions argument for stability of solitons,this dissertation proves the existence of small solitons,then a new cross constrained variational problem is introduced and solved using the profile decomposition argument.The combination of two types of variational problems proves the orbital stability of small solitons and the correspondence between the the prescribed mass and soliton frequency.Finally,by using the bootstrap argument and uniform estimates,this dissertation constructs multi solitons with different speeds of the nonlinear Schr(?)dinger equation with double power nonlinearity.2.This dissertation studies the existence of high speed excited multi solitons for the competitive power nonlinear Schr(?)dinger equation.Firstly,by using interpolation inequality,Young inequality,energy and mass conservation,this dissertation proves the global well posedness of the competitive power nonlinear Schr(?)dinger equation.Then the exponential decay property of excited solitons is given,and the coercivity of excited multi solitons is proved by combining some spectral properties.Finally,the existence of high speed excited multi solitons is proved by uniform estimates and bootstrap argument.3.This dissertation studies the existence and orbital stability of small solitons for generalized Davey-Stewartson system in two dimensional space,the essential correspondence between the prescribed mass and soliton frequency in the normalized solution problem corresponding to small solitons,and the multi soliton properties of equations are constructed by stable small solitons.Firstly,this dissertation proves the global well posedness of the generalized Davey-Stewartson system in two dimensional space.Secondly,by using the profile decomposition argument to solve the Cazenave-Lions variational problem,this dissertation proves the existence of small solitons.Then a new cross constrained variational problem is introduced and solved.The combination of two types of variational problems proves the orbital stability of small solitons and the correspondence between the the prescribed mass and soliton frequency,meanwhile,this dissertation gives some spectrum results.Finally,the existence of multi-solitons constructed from stable small solitons is proved by bootstrap argument and uniform estimates.