Exponential time differencing Crank–Nicolson method with a quartic spline approximation for nonlinear Schrödinger equations

Abstract

This paper studies a central difference and quartic spline approximation based exponential time differencing Crank–Nicolson (ETD-CN) method for solving systems of one-dimensional nonlinear Schrödinger equations and two-dimensional nonlinear Schrödinger equations. A local extrapolation is employed to achieve a fourth order accuracy in time. The numerical method is proven to be highly efficient and stable for long-range soliton computations. Numerical examples associated with Dirichlet, Neumann and periodic boundary conditions are provided to illustrate the accuracy, efficiency and stability of the method proposed.