Numerical Solution Of Nonlinear Schrodinger Equations On Unbounded Domains

The numerical solution of the nonlinear Schrodinger equations on unbounded domain(including the logarithmic Schrodinger equation and the nonlinear Schrodinger equation with wave operator)is considered in this paper.The nonlinear Schrodinger equations are widely used in many important areas,such as atomic physics,nuclear physics,and solid state physics.Due to the blow up of the nonlinear logarithmic term,a regularized version of the logarithmic Schrodinger equation on unbounded domains with a small regularization parameter is developed.It is difficult to de-velop efficient numerical methods for the nonlinear Schrodinger equation(including the regularized logarithmic Schrodinger equation and the nonlinear Schrodinger e-quation with wave operator)on unbounded domains,since the physical domain is unbounded and the equations are nonlinear.In order to overcome the unbound-edness of the physical domain,the artificial boundary method is adopted,and the appropriate artificial boundary conditions are designed in the introduced artificial boundaries.Based on the idea of the well-known operator splitting method,the unified approach is applied to overcome the nonlinearity of the Schrodinger equa-tions,and the local artificial boundary conditions are obtained.Then,the original nonlinear problem on unbounded domain is reduced to an initial boundary value problem on a bounded domain,which can be efficiently solved by the finite dif-ference method.The convergence and the stability of the reduced problem on the bounded domain are analyzed by introducing some auxiliary variables.Numerical results are reported to verify the accuracy and effectiveness of our proposed method.