The Study Of The Search Extension Method For Solving Nonlinear Schr(?)dinger Eigenvalue Problems And Its Applications

The nonlinear Schr(?)dinger eigenvalue problem is widely used in modern sciences such as nonlinear optics,plasma ion acoustic waves,and electronic structure calculation.However,due to the nonlinearity of the model,the multiplicity and instability of the solution,the numerical calculation is quite challenging.In this paper,the improved search extension method(SEM)is designed to solve the nonlinear Schr(?)dinger eigenvalue problem,and a stable algorithm is used to calculate multiple unstable solutions.Firstly,the method uses the linear combination of the eigen-bases of the model problem corresponding to the linear eigenvalue problem to search for the initial value of the multiple solutions.Then,based on the existing eigen-bases,the eigen-bases function is appropriately added,and the linear combination of the new eigen-bases is used to search for better initial value.Then,combining the interpolation coefficient technique and the Legendre-Galerkin spectral method discrete model problem,a simple form of nonlinear algebraic equations is obtained.Therefore,the Jacobian matrix of each Newton iteration only needs to update a relatively simple matrix.Finally,the numerical extension method is used to solve the solution corresponding to the given initial value.The algorithm is computationally intensive and has spectral accuracy.In this paper,it is applied to the calculation of the ground state and excited state of Bose-Einstein condensate(BEC),which effectively simulates the Bose-Einstein condensed ground state and multiple excited state under different potential functions and parameters.Some interesting properties of the ground state and the excited state solution are found.