Analysis of a Crank-Nicolson finite difference scheme for (2 + 1)D perturbed nonlinear Schrödinger equations with saturable nonlinearity.

We analyze a Crank–Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schrödinger equation with saturable nonlinearity and a perturbation of cubic loss. We show the boundedness, the existence and uniqueness of a numerical solution. We establish the error bound to prove the convergence of the numerical solution. Moreover, we find that the convergence rate is at the second order in both time step and spatial mesh size only under a mild and simple assumption. The numerical scheme is validated by the extensive simulations of the (2+1)D saturable nonlinear Schrödinger equation with cubic loss. The simulations for traveling 2D solitons are implemented by using an accelerated imaginary-time evolution scheme and the Crank–Nicolson finite difference method. • The (2+1)D NLS saturable equations can stabilize 2D solitons. • The rigorous analysis for the perturbed (2+1)D saturable NLS equation is demonstrated for the first time. • By appropriate settings, the existence and the uniqueness of a solution are proved. • An error analysis is established and validated by the simulations for traveling 2D solitons. [ABSTRACT FROM AUTHOR]