A new algorithm for solving one-dimensional Schrödinger equations in the reproducing kernel space

On the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear Schrödinger equations is presented. The analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. The convergence of the approximate solution to the analytical solution is also proved. The method is examined for the single soliton solution and interaction of two solitons. Numerical experiments show that the proposed method is of satisfactory accuracy and preserves the conservation laws of charge and energy. The numerical results are compared with both the analytical and numerical solutions of some earlier papers in the literature.