A generalized finite-difference time-domain scheme for solving nonlinear Schrödinger equations

Recently, we have developed a generalized finite-difference time-domain (G-FDTD) method for solving the time dependent linear Schrödinger equation. The G-FDTD is explicit and permits an accurate solution with simple computation, and also relaxes the stability condition as compared with the original FDTD scheme. In this article, we extend the G-FDTD scheme to solve nonlinear Schrödinger equations. Using the discrete energy method, the G-FDTD scheme is shown to satisfy a discrete analogous form of the conservation law. The obtained scheme is tested by three examples of soliton propagation, including bright and dark solitons as well as a 2D case. Compared with other popular existing methods, numerical results show that the present scheme provides a more accurate solution.