Efficient time-splitting Hermite-Galerkin spectral method for the coupled nonlinear Schrödinger equations

The paper focuses on efficient time-splitting Hermite-Galerkin spectral approximation of the coupled nonlinear Schrödinger equations on the whole line. The original problem is decomposed into one nonlinear subproblem and one linear subproblem by time-splitting method. At each time step, the nonlinear subproblem is solved exactly. While the linear subproblem is efficiently solved by choosing suitable Hermite basis functions with matrix decomposition technique. Numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed method.