The Study Of Numerical Method Based On Bose-Einstein Condensate

The Bose–Einstein condensate is a gaseous,superfluid matter state of a boson atom that cools to near absolute zero.In this state,almost all atoms are concentrated in the quantum state with the lowest energy,forming a macroscopic quantum state.The theoretical and numerical methods for BEC have attracted the attention of scholars.Computation the ground state,first excited state and dynamic properties of BEC is one of the basic problems of BEC research.The classical nonlinear Schr?dinger(NLS)equation,also known as the Gross-Pitaevskii equation(GPE),has been widely used to describe the ground state and dynamics of BECs.This paper studies the fractional Bose-Einstein condensates with three different potential wells.We choose the weighted shift Grünwald-Letnikov difference method(WSGD)and the compact implicit integral factor method for BECs.We use the second-order central difference method and Krylov subspace to solve the spin-orbit coupling Bose-Einstein condensates.The properties of energy decline and the feasibility are verified by the numerical analysis and numerical experiments.The first chapter introduces the basic concept of Bose-Einstein condensate and describes the development history and practical application.The second chapter discuss the three numerical methods and discusses the accuracy,efficiency,and stability of each numerical method.In the third chapter,the Bose-Einstein condensate of fractional Laplacian with three different potential wells is studied.Firstly,The ground state problem is transformed into the minimum energy problem for solving the fractional-order Gross-Pitaevskii equation by the normalized gradient flow method.The weighted shift Grünwald-Letnikov difference method(WSGD)with weighted offset is used for spatial discretization.The discrete result is a set of ordinary differential equations with second-order precision and unconditionally stable.The time discretization adopts the integration factor method,which has the feature of high calculation precision,small storage capacity and high efficiency.The fourth chapter discusses the spin-orbit coupling Bose-Einstein condensate state(SOC BEC).By using the finite difference method,the semi-discrete system of ordinary differential equations is obtained.In order to effectively estimate the matrix exponential operator,the Krylov subspace approximation is applied to the matrix exponential operator,which has high computational efficiency,small CPU occupation and strong stability.Numerical experiments show the ground state of SOC BEC and compare the effects of different parameters on the ground state density of SOC BEC.