Research On Numerical Methods Of Some Nonlinear Schr?dinger Equations

The nonlinear Schr?dinger equation is a basic equation with strong universality in nonlinear science.This equation has a wide range of applications in many fields,such as nonlinear optics,quantum mechanics,fluid mechanics,plasma physics,and dynamics of Bose-Einstein condensates at extremely low temperatures.With the development of computing technology,more and more types of nonlinear Schr?dinger equations have been studied by many scholars,and rich and diverse numerical methods have been proposed.This paper mainly studies numerical methods to solve three types of nonlinear Schr?dinger equations: the nonlinear Schr?dinger equation in the plane rectangular coordinate system,the polar coordinate system,and the two-dimensional fractional nonlinear Schr?dinger equation.A conservative finite difference scheme is proposed for the two-dimensional nonlinear Schr?dinger equation in plane Cartesian coordinate system.In order to improve computational efficiency,we first write the difference matrix of the two-dimensional Laplacian operator in the form of Kronecker product.Then the properties of Kronecker product are applied to diagonalize the differential matrix to obtain the corresponding eigenvalues and eigenvectors.Finally,the fast Fourier transform is used to realize the fast solving.Numerical experiments verifies the conservation of discrete mass and energy.It also greatly reduced computation storage and CPU time,which confirmed the efficiency of this method.For the two-dimensional nonlinear Schr?dinger equation in the polar coordinates,we first write the Laplacian operator in polar coordinates format,and then the computation domain is divided in the r direction and ? direction.The centered finite is used for spatial discretization and the integration factor method is applied for the time discretization.Then we apply Kronecker product and matrix vectorization to solve the two-dimensional nonlinear Schr?dinger equation in polar coordinates.In the implementation process,the Kroylov subspace method is used to solve the product of the exponential matrix and the vector,and finally the fixed point iteration method is used to solve the nonlinear equations.The results of numerical experiments show that this method can capture blow-up solution well.For the two-dimensional fractional nonlinear Schr?dinger equation,the integral derivative is firstly extended to the fractional derivative.The fractional power of the spectral differential matrix is obtained by eigen-decomposition and interpolation approximation,and the differential matrix of the fractional derivative is obtained.Then,we prove the conservation of mass and energy in the semi-discretized scheme.The method of compact integration factor is used to discretize in time direction.Finally,the conservation of mass and energy is verified by numerical experiments,and it is also shown that the proposed algorithm can capture the blow-up solution while maintaining the high precision.The first chapter of this article introduces the concept and research status of nonlinear Schr?dinger equation,and elaborates the innovation of this article.The second chapter explores the derivation process of the differential matrix,the derivation process of the eigenvalues and eigenvectors of the differential matrix,and the concepts and properties applied in subsequent articles.The third chapter discusses the numerical method of the nonlinear Schr?dinger equation in the Cartesian coordinate system.Chapter 4 gives the numerical simulation for the nonlinear Schr?dinger equation in polar coordinates.Chapter 5 extends the integer order to the fractional order,and studies the numerical simulation of the fractional-order nonlinear Schr?dinger equation.