Study On The Problems Related To The Damped Stochastic Nonlinear Schr(?)dinger Equations

With the development of science and technology and the cross-integration between various disciplines,stochastic partial differential equations have gradually become an important mathematical tool for the qualitative and quantitative study of stochastic problems,and have received wide attention.As a special class of stochastic partial differential equations,the stochastic nonlinear Schr(?)dinger equations are widely used in quantum mechanics,communication optical fiber,etc.This thesis focuses on the existence and uniqueness of solution of damped stochastic nonlinear Schr(?)dinger equation with quadratic potential,and constructs numerical schemes to preserve the stochastic conformal multi-symplectic structure of damped stochastic nonlinear Schr(?)dinger equation driven by additive noise.Specifically,first,we study the global well-posedness of damped stochastic nonlinear Schr(?)dinger equation with quadratic potential with the help of the fixed point theorem and energy functional;second,we give the evolution laws of the charge and energy of the damped stochastic nonlinear Schr(?)dinger equation solution driven by additive noise,and the stochastic conformal Preissman and stochastic conformal Euler box schemes are constructed to preserve the stochastic conformal multi-symplectic geometry of the original equation.This thesis is divided into four parts,and the overall structure is as follows:The first chapter is about the research background of the damped stochastic nonlinear Schr(?)dinger equations and the current research situation at home and abroad,and introduces the main content and results of this thesis.The second chapter is preparatory knowledge,gives several inequalities often used in the thesis,and introduces the Wiener process and random integral in Hilbert space to lay the theoretical foundation for later research.The third chapter on the global well-posedness of the damped stochastic nonlinear Schr(?)dinger equation with quadratic potential.First,we give the local well-posedness of the equation with the help of the fixed point theorem,and then obtain the global well-posedness by proving the consistent boundedness of the energy functional.The fourth chapter focuses on the stochastic conformal multi-symplectic schemes of the damped stochastic nonlinear Schr(?)dinger equation.First,we use the It(?) process to give the evolution laws of the charge and energy of the equation,and construct stochastic conformal Preissman schemes and stochastic conformal Euler box schemes that preserve the stochastic conformal multi-symplectic geometry of the original equation.Furthermore,we give fully discrete forms of the charge and energy evolution laws in the stochastic conformal Preissman scheme.