A Crank-Nicolson finite difference scheme for coupled nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping

Abstract

We propose a Crank-Nicolson finite difference scheme to simulate a 2D perturbed soliton interaction under the framework of coupled (2+1)D nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping. We rigorously demonstrate that the proposed numerical scheme achieves a second-order convergence rate in both the discrete $H_0^1$ and $L^2$ norms, relative to the time step and spatial mesh size. We establish the boundedness of discrete energies to prove the existence and uniqueness of the solutions derived from the Crank-Nicolson scheme. The validity of the analysis is confirmed through numerical simulations that apply to the corresponding coupled (2+1)D saturable nonlinear Schrödinger equations with damping terms.