Time-splitting pseudo-spectral domain decomposition method for the soliton solutions of the one- and multi-dimensional nonlinear Schrödinger equations

In this paper, we study the simulation of nonlinear Schrödinger equation in one, two and three dimensions. The proposed method is based on a time-splitting method that decomposes the original problem into two parts, a linear equation and a nonlinear equation. The linear equation in one dimension is approximated with the Chebyshev pseudo-spectral collocation method in space variable and the Crank–Nicolson method in time; while the nonlinear equation with constant coefficients can be solved exactly. As the goal of the present paper is to study the nonlinear Schrödinger equation in the large finite domain, we propose a domain decomposition method. In comparison with the single-domain, the multi-domain methods can produce a sparse differentiation matrix with fewer memory space and less computations. In this study, we choose an overlapping multi-domain scheme. By applying the alternating direction implicit technique, we extend this efficient method to solve the nonlinear Schrödinger equation both in two and three dimensions, while for the solution at each time step, it only needs to solve a sequence of linear partial differential equations in one dimension, respectively. Several examples for one- and multi-dimensional nonlinear Schrödinger equations are presented to demonstrate high accuracy and capability of the proposed method. Some numerical experiments are reported which show that this scheme preserves the conservation laws of charge and energy.