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We analyze a Crank-Nicolson finite difference discretization for theperturbed (2+1)D nonlinear Schr\"odinger equation with saturable nonlinearityand a perturbation of cubic loss. We show the boundedness, the existence anduniqueness of a numerical solution. We establish the error bound to prove theconvergence of the numerical solution. Moreover, we find that the convergencerate is at the second order in both time step and spatial mesh size under amild assumption. The numerical scheme is validated by the extensive simulationsof the (2+1)D saturable nonlinear Schr\"odinger model with cubic loss. Thesimulations for travelling solitons are implemented by using an acceleratedimaginary-time evolution scheme and the Crank-Nicolson finite differencemethod.

Analysis of a Crank-Nicolson finite difference scheme for (2+1)D perturbed nonlinear Schr\"odinger equations with saturable nonlinearity