High-precision Numerical Solutions Of Nonlinear Schr?dinger Equations

The paper focuses on the high-order time-splitting Hermite-Galerkin spectral methods for two nonlinear Schr?dinger equations on the whole line.For the coupled nonlinear Schr?dinger equation on the whole line,it is first split into two subproblems: nonlinear subproblem and linear subproblem.The nonlinear subproblem can be solved exactly.By the orthogonal decomposition of matrix,appropriate Hermite basis functions are constructed.Some appropriate function transformation leads to diagonal matrices.The equations is then efficiently solved by combining second-order time-splitting and fourth-order time-splitting schemes.Time-splitting Hermite-Galerkin spectral schemes with scaling factors are then considered.Finally,some numerical examples are presented.The numerical results demonstrate that the proposed schemes have high-order convergence rate on both time and space and preserve the discrete mass invariant.For the coupled nonlinear fractional Schr?dinger equation,it is split into two subproblems similarly.By constructing appropriate basis functions and combing with the time-splitting method,the equation can also be efficiently solved.Numerical experiments show again that the proposed schemes have high-order convergence rate on both time and space and preserve the discrete mass invariant.