Uniform error estimates of a Crank-Nicolson finite difference scheme for nonlinear Schrödinger equations with cubic gain/loss and quintic loss

We use the standard Crank-Nicolson finite difference scheme to discretize the (2+1)D cubic nonlinear Schrödinger equation with cubic gain/loss and quintic loss. By analyzing the discrete L 2 and 𝐻 1 0 norms of the numerical solutions, we demonstrate the existence and uniqueness of the discrete solution. Additionally, error estimates are derived, showing second-order accuracy in both space and time with respect to the discrete L 2 and 𝐻 1 0 norms.