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Solitons of the purely cubic nonlinear Schrödinger equation in a space dimension of n ≥ 2 suffer critical and supercritical collapses. These solitons can be stabilized in a cubic-quintic nonlinear medium. In this paper, we analyze the Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping. We show that both the discrete solution, in the discrete L 2 -norm, and discrete energy are bounded. By using appropriate settings and estimations, the existence and the uniqueness of the numerical solution are proved. In addition, the error estimations are established in terms of second order for both space and time in discrete L 2 -norm and H 1 -norm. Numerical simulations for the (2+1)D cubic-quintic nonlinear Schrödinger equation with cubic damping are conducted to validate the convergence. • The (2+1)D cubic-quintic NLS equation can against 2D-soliton collapses. • A Crank-Nicolson finite difference scheme for the damped (2+1)D cubic-quintic NLS equation is proposed. • The existence and uniqueness of solutions are demonstrated using the boundedness of discrete energy and mass. • The scheme converges second order in time and space. • The numerical experiments are demonstrated to verify the theoretical results. [ABSTRACT FROM AUTHOR]

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]