A Crank-Nicolson Finite Difference Scheme for the (2+1)D Saturable Nonlinear Schrödinger Equation with Generalized Damping

Abstract

In this study, we implement a Crank-Nicolson finite difference scheme to discretize the (2+1)D saturable nonlinear Schrödinger equation with general damping. We show the existence and uniqueness of the discrete solution. The boundedness of the discrete mass and energy is established. The error between the exact and discrete solutions is shown to converge at a second-order rate in both time and space, according to the $L^2$ and $H^1$ discrete norms. Moreover, we show that the proposed scheme preserves the mass conservation and energy conservation for the (2+1)D saturable nonlinear Schrödinger equation without damping. Numerical simulations are conducted to validate these convergence properties and the conservation laws.