Numerical solution of coupled nonlinear Schrödinger equations on unbounded domains

Abstract

In this paper, we investigate the numerical solution of general N-coupled nonlinear Schrödinger equations on unbounded domains, that describe the multiple solitary waves in physics. Due to the unboundedness of the physical domain and strong nonlinearity, how to design appropriate numerical approach to obtain the numerical solution and simulate the real physical phenomena is a major challenge. Based on the ideas of artificial boundary method and operator splitting approach, we develop local artificial boundary conditions for the N-coupled nonlinear Schrödinger equations on the introduced artificial boundaries. Then the original initial value problem was reduced into an initial boundary value problem on the truncated computational domain, which can be solved by the finite difference method efficiently. We rigorously prove the stability of the reduced problem by introducing a series of auxiliary variables, which are vital to overcome the difficulty of the mixed partial derivatives in the local artificial boundary conditions. Numerical simulations are also presented to validate accuracy and the stability of our proposed method.