A Finite Difference Scheme for (2+1)D Cubic-Quintic Nonlinear Schr\"odinger Equations with Nonlinear Damping

TL;DR

This paper develops and analyzes a finite difference numerical scheme for the (2+1)D cubic-quintic nonlinear Schrödinger equation with damping, proving stability, convergence, and validating results through simulations.

Contribution

It introduces a Crank-Nicolson finite difference scheme for the (2+1)D cubic-quintic NLS with damping, establishing stability, existence, uniqueness, and second-order error estimates.

Findings

Discrete solution and energy are bounded in $L^2$ norm.

Existence and uniqueness of the numerical solution are proved.

Numerical simulations confirm convergence and stability.

Abstract

Solitons of the purely cubic nonlinear Schr\"odinger equation in a space dimension of $n \geq 2$ 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 Schr\"odinger 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 Schr\"odinger equation with cubic damping are conducted to validate the convergence.